# Divisor-Bounded Multiplicative Functions in Short Intervals

Alexander P. Mangerel  Centre de Recherches Mathématiques, Université de
Montréal, Montréal, Québec<EMAIL_ADDRESS>
###### Abstract.

We extend the Matomäki-Radziwiłł theorem to a large collection of unbounded
multiplicative functions that are uniformly bounded, but not necessarily
bounded by 1, on the primes. Our result allows us to estimate averages of such
a function $f$ in typical intervals of length $h(\log X)^{c}$, with
$h=h(X)\rightarrow\infty$ and where $c=c_{f}\geq 0$ is determined by the
distribution of $\\{|f(p)|\\}_{p}$ in an explicit way. We give three
applications.
First, we show that the classical Rankin-Selberg-type asymptotic formula for
partial sums of $|\lambda_{f}(n)|^{2}$, where $\\{\lambda_{f}(n)\\}_{n}$ is
the sequence of normalized Fourier coefficients of a primitive non-CM
holomorphic cusp form, persists in typical short intervals of length $h\log
X$, if $h=h(X)\rightarrow\infty$. We also generalize this result to sequences
$\\{|\lambda_{\pi}(n)|^{2}\\}_{n}$, where $\lambda_{\pi}(n)$ is the $n$th
coefficient of the standard $L$-function of an automorphic representation
$\pi$ with unitary central character for $GL_{m}$, $m\geq 2$, provided $\pi$
satisfies the generalized Ramanujan conjecture.
Second, using recent developments in the theory of automorphic forms we
estimate the variance of averages of all positive real moments
$\\{|\lambda_{f}(n)|^{\alpha}\\}_{n}$ over intervals of length $h(\log
X)^{c_{\alpha}}$, with $c_{\alpha}>0$ explicit, for any $\alpha>0$, as
$h=h(X)\rightarrow\infty$.
Finally, we show that the (non-multiplicative) Hooley $\Delta$-function has
average value $\gg\log\log X$ in typical short intervals of length $(\log
X)^{1/2+\eta}$, where $\eta>0$ is fixed.

##  1\. Introduction and Main Results

###  1.1. The Matomäki-Radziwiłł theorem for bounded multiplicative functions

The Matomäki-Radziwiłł theorem, in its various incarnations, gives estimates
for the error term in approximating the average of a bounded multiplicative
function in a typical short interval by a corresponding long interval average.
In the breakthrough paper [26], the authors showed that, uniformly over all
real-valued multiplicative functions $f:\mathbb{N}\rightarrow[-1,1]$, for any
$1\leq h\leq X$ such that $h=h(X)\rightarrow\infty$ as $X\rightarrow\infty$,

(1) $\frac{1}{h}\sum_{x-h<n\leq x}f(n)=\frac{2}{X}\sum_{X/2<n\leq X}f(n)+o(1)$

for all but $o(X)$ integers $x\in[X/2,X]$. A key feature of this result is
that the interval length $h$ can grow arbitrarily slowly as a function of $X$.
This result has had countless applications to a variety of problems across
mathematics, including to partial results towards Chowla’s conjecture on
correlations of the Liouville function [37], [39], the resolution of the
famous Erdős discrepancy problem [36], and progress on Sarnak’s Möbius
disjointness conjecture (e.g., [38], [3]; see [20] for a more exhaustive
list).

Since [26], the result has been extended and generalized in various
directions. In [27], a corresponding short interval result was given for non-
pretentious complex-valued multiplicative functions
$f:\mathbb{N}\rightarrow\mathbb{U}$, where
$\mathbb{U}:=\\{z\in\mathbb{C}:|z|\leq 1\\}$. To be precise, if we define

$D_{f}(X;T):=\min_{|t|\leq T}\mathbb{D}(f,n^{it};X)^{2}:=\min_{|t|\leq
T}\sum_{p\leq X}\frac{1-\text{Re}(f(p)p^{-it})}{p},$

where $\mathbb{D}$ denotes the Granville-Soundararajan pretentious distance,
they showed that if $f:\mathbb{N}\rightarrow\mathbb{U}$ satisfies
$D_{f}(X;X)\rightarrow\infty$ then

$\left|\frac{1}{h}\sum_{x-h<n\leq x}f(n)\right|=o(1)$

for all but $o(X)$ integers $x\in[X/2,X]$, whenever $h=h(X)\rightarrow\infty$.
In a different direction, exploring the heuristic relationship between the
distributions of arithmetic functions in short intervals and in short
arithmetic progressions, Klurman, the author and Teräväinen [19] obtained an
analogue of (1) for typical111Complications arise concerning both the prime
divisors of the modulus $q$ as well as the distribution of zeros of Dirichlet
$L$-functions $\pmod{q}$, so the theorem proven in [19] is qualitatively
weaker than (1) unconditionally in general. short arithmetic progressions.

In the recent paper [24], a widely generalized version of the results of [26]
was developed, which among other things extended the work of [27]. The authors
showed that for a general complex-valued multiplicative function
$f:\mathbb{N}\rightarrow\mathbb{U}$, if $t_{0}=t_{0}(f,X)$ is a minimizer in
the definition of $D_{f}(X;X)$ then for all but $o(X)$ integers $x\in[X/2,X]$,
one obtains an asymptotic formula with main term of the form

(2) $\frac{1}{h}\sum_{x-h<n\leq
x}f(n)=\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}+o(1),$

with a better quantitative dependence of the bound for the exceptional set on
the interval length $h$ than in [26].

By Shiu’s theorem (Lemma 3.2 below), we have

$\frac{1}{X}\sum_{n\leq X}|f(n)|\ll\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right),$

so (2) is trivial whenever $\sum_{p\leq
X}\frac{1-|f(p)|}{p}\rightarrow\infty$, for instance if $f(p)=0$ significantly
often on the primes. Rectifying this weakness, Matomäki and Radziwiłł improved
the quality of the $o(1)$ error term for a large collection of $1$-bounded
functions with _sparse_ prime support. Specifically, they showed that if there
are constants $A>0$ and $\theta\in(0,1]$ such that the sieve-type lower bound
condition

(3) $\sum_{z<p\leq w}\frac{|f(p)|}{p}\geq A\sum_{z<p\leq
w}\frac{1}{p}-O\left(\frac{1}{\log z}\right)\text{ holds for all }2\leq z\leq
w\leq X^{\theta}$

then one can improve the $o(1)$ term to

$o\left(\prod_{p\leq X}\left(1+\frac{|f(p)|-1}{p}\right)\right).$

This savings comes at a natural cost, namely that the length of the interval
$h$ is no longer arbitrarily slow growing as a function of $X$, but must grow
in a manner that depends on the sparseness of the support222As pointed out in
[24, p. 8], it is generally unclear what the least size of such intervals must
be for a given bounded multiplicative function. of $f$.

Precisely, the main result of [24] may be stated as follows. In the sequel,
for a multiplicative function $f:\mathbb{N}\rightarrow\mathbb{U}$ we write

$H(f;X):=\prod_{p\leq X}\left(1+\frac{(|f(p)|-1)^{2}}{p}\right).$

######  Theorem (Matomäki-Radziwiłł, [24] Thm. 1.9).

Let $A>0$ and $\theta\in(0,1]$. Let $f:\mathbb{N}\rightarrow\mathbb{U}$ be a
multiplicative function such that (3) holds for all $2\leq z\leq w\leq
X^{\theta}$. Let $2\leq h_{0}\leq X^{\theta}$ and put $h:=h_{0}H(f;X)$. Also,
set $t_{0}=t_{0}(f,X)$. Then there are constants333In [24] they obtained the
explicit constant $\rho_{A}=A/3-\frac{2}{3\pi}\sin(\pi A/2)$. $C=C(\theta)>1$,
$\rho_{A}>0$ such that for any $\delta\in(0,1/1000)$ and $0<\rho<\rho_{A}$,

$\displaystyle\left|\frac{1}{h}\sum_{x-h<n\leq
x}f(n)-\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}\right|$ $\displaystyle\leq\left(\delta+C\left(\frac{\log\log
h_{0}}{\log h_{0}}\right)^{A}+(\log X)^{-A\rho/36}\right)\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right),$

for all $x\in[X/2,X]$ outside of a set of size

$\ll_{\theta}X\left(h^{-(\delta/2000)^{1/A}}+X^{-\theta^{3}(\delta/2000)^{6/A}}\right).$

###  1.2. Divisor-bounded multiplicative functions

Let $B\geq 1$. We define the _generalized $B$-divisor function_ $d_{B}(n)$ via

$\zeta(s)^{B}=\sum_{n\geq 1}\frac{d_{B}(n)}{n^{s}}\text{ for }\text{Re}(s)>1.$

It can be deduced that $d_{B}(n)$ is multiplicative, and moreover
$d_{B}(p^{k})=\binom{B+k-1}{k}$, for all $k\geq 1$. In particular,
$d_{B}(p)=B$. For integer values of $B$ this coincides with the usual $B$-fold
divisor functions, e.g., when $B=2$ we have $d_{B}(n)=d(n)$, and when $B=1$ we
have $d_{B}(n)\equiv 1$.

We say that a multiplicative function $f:\mathbb{N}\rightarrow\mathbb{C}$ is
_divisor-bounded_ if there is a $B\geq 1$ such that $|f(n)|\leq d_{B}(n)$ for
all $n$. When $B=2$, for example, this includes functions such as the twisted
divisor function $d(n,\theta):=\sum_{d|n}d^{i\theta}$ for
$\theta\in\mathbb{R}$, as well as $r(n)/4$, where
$r(n):=|\\{(a,b)\in\mathbb{Z}^{2}:a^{2}+b^{2}=n\\}|$.

There is a rich literature about mean values of general, 1-bounded
multiplicative functions. The works of Wirsing [45] and Halász [10] are
fundamental, with noteworthy developments by Montgomery [29] and Tenenbaum
[41, Thm. III.4.7]. The theory has recently undergone an important change in
perspective, due in large part to the extensive, pioneering works of Granville
and Soundararajan (e.g., [8], [9]). This well-formed theory significantly
informs the results of [26] and [24].

In comparison, the study of long averages of general _unbounded_
multiplicative functions has only garnered significant interest more recently.
Granville, Harper and Soundararajan [7], in developing a new proof of a
quantitative form of Halász’ theorem, were able to obtain bounds for averages
of multiplicative functions $f:\mathbb{N}\rightarrow\mathbb{C}$ for which the
coefficients of the Dirichlet series444Implicitly, it is assumed that
$-\frac{L^{\prime}}{L}(s,f)$ is well-defined in $\text{Re}(s)>1$.

(4) $-\frac{L^{\prime}}{L}(s,f)=\sum_{n\geq
1}\frac{\Lambda_{f}(n)}{n^{s}},\text{ where }L(s,f):=\sum_{n\geq
1}\frac{f(n)}{n^{s}}\text{ for }\text{Re}(s)>1,$

satisfy the bound $|\Lambda_{f}(n)|\leq B\Lambda(n)$ uniformly over
$n\in\mathbb{N}$ for some $B\geq 1$, where $\Lambda(n)$ is the von Mangoldt
function. Such functions satisfy $|f(n)|\leq d_{B}(n)$ for all $n$. In [42],
Tenenbaum, improving on qualitative results due to Elliott [1], established
quantitative upper bounds and asymptotic formulae for the ratios $|\sum_{n\leq
X}f(n)|/(\sum_{n\leq X}|f(n)|)$, assuming $f$ is uniformly bounded on the
primes, not too large on average at prime powers, and satisfies a hypothesis
like (3). See also [23, Ch. 2] for results of a similar kind under stronger
hypotheses.

On the basis of these developments, it is reasonable to ask whether the
results of [26] and [24] can be extended to divisor-bounded functions of a
certain type. This was hinted at in [24, p. 9] but, as far as the author is
aware, it does not yet exist in the literature. The purpose of this paper is
to establish such extensions for a broad class of divisor-bounded
multiplicative functions, among other unbounded functions.

In the following subsection we provide three examples that motivate our main
theorem, Theorem 1.7. Besides the applications we give here, this result will
also be applied in [22] to study short interval averages of general additive
functions.

###  1.3. Applications

####  1.3.1. Rankin-Selberg estimates for $GL_{m}$ in typical short intervals

Let $f$ be a fixed even weight $k\geq 2$, level $1$ primitive, Hecke-
normalized holomorphic cusp form without complex multiplication, and write its
Fourier expansion at $\infty$ as

$f(z)=\sum_{n\geq 1}\lambda_{f}(n)n^{\frac{k-1}{2}}e(nz),\quad\text{Im}(z)>0,$

with $\lambda_{f}(1)=1$. Set also

$g_{f}(n):=\sum_{d^{2}|n}|\lambda_{f}(n/d^{2})|^{2}.$

By the Hecke relation

$\lambda_{f}(m)\lambda_{f}(n)=\sum_{d|(m,n)}\lambda_{f}\left(\frac{mn}{d^{2}}\right),\quad
m,n\in\mathbb{N},$

$|\lambda_{f}|^{2}$ and thus also $g_{f}$ are multiplicative functions.
Deligne showed that $|\lambda_{f}(p)|\leq 2$ for all primes $p$, and in
general $|\lambda_{f}(n)|^{2}\leq d(n)^{2}$. Thus $|\lambda_{f}|^{2}$ is
bounded by a power of a divisor-function, and the same can be shown for
$g_{f}$.

The classical Rankin-Selberg method shows [18, Sec. 14.9] that asymptotic
formulae

$\displaystyle\frac{1}{X}\sum_{n\leq
X}|\lambda_{f}(n)|^{2}=c_{f}+O(X^{-2/5}),\quad\quad\frac{1}{X}\sum_{n\leq
X}g_{f}(n)=d_{f}+O(X^{-2/5}),$

hold as $X\rightarrow\infty$, where $c_{f},d_{f}>0$ are constants depending on
$f$. The Rankin-Selberg problem is equivalent to asking for an improvement of
the error term $X^{-2/5}$ in both of these estimates, but this is not our
point of interest here.

One can ask whether the above asymptotic formulae continue to hold in short
intervals. Ivić [17] considered the variance of the error term in short
interval averages. Specifically, he showed [17, Cor. 2] on the Lindelöf
hypothesis that

(5) $\frac{1}{X}\int_{X}^{2X}\left(\frac{1}{h}\sum_{x<n\leq
x+h}g_{f}(n)-d_{f}\right)^{2}dx=o(1),$

as long as $h\geq X^{2/5-\varepsilon}$, albeit with a power-saving error term
in the latter range.
At the expense of the quality of the error term, we obtain the following
improvement in the range where (5) holds.

######  Corollary 1.1.

Let $10\leq h_{0}\leq X/(10\log X)$ and set $h:=h_{0}\log X$. Then there is a
constant $\theta>0$ such that

$\frac{1}{X}\int_{X}^{2X}\left(\frac{1}{h}\sum_{x<n\leq
x+h}|\lambda_{f}(n)|^{2}-c_{f}\right)^{2}dx\ll\frac{\log\log h_{0}}{\log
h_{0}}+\frac{\log\log X}{(\log X)^{\theta}}.$

The same estimate holds when $|\lambda_{f}|^{2}$ and $c_{f}$ are replaced by
$g_{f}$ and $d_{f}$, respectively.

This corollary might appear surprising, given our currently incomplete
understanding of the shifted convolution problem

$\sum_{X<n\leq 2X}|\lambda_{f}(n)|^{2}|\lambda_{f}(n+r)|^{2},\quad\quad
1\leq|r|\leq h.$

Actually, our proof of Corollary 1.1 relies only on the multiplicativity of
$|\lambda_{f}|^{2}$, Deligne’s theorem and the prime number theorem for
Rankin-Selberg $L$-functions (see e.g., Lemma 5.6). This suggests555We would
like to thank Maksym Radziwiłł and Jesse Thorner for pointing this out. that a
generalization to coefficients of automorphic $L$-functions for $GL_{n}$
should be possible, provided that these satisfy the generalized Ramanujan
conjecture and hence are divisor-bounded.
To be more precise, let $m\geq 2$, let $\mathbb{A}$ be the ring of adeles of
$\mathbb{Q}$, and let $\pi$ be a cuspidal automorphic representation of
$GL_{m}(\mathbb{A})$ with unitary central character that acts trivially on the
diagonally embedded copy of $\mathbb{R}^{+}$. We let $q_{\pi}$ denote the
conductor of $\pi$. The finite part of $\pi$ factors as a tensor product
$\pi=\otimes_{p}\pi_{p}$, with local representations $\pi_{p}$ at each prime
$p$. The local $L$-function at $p$ takes the form

$L(s,\pi_{p})=\prod_{1\leq j\leq
m}\left(1-\frac{\alpha_{j,\pi}(p)}{p^{s}}\right)^{-1}:=\sum_{l\geq
0}\frac{\lambda_{\pi}(p^{l})}{p^{ls}},$

where $\\{\alpha_{1,\pi}(p),\ldots,\alpha_{m,\pi}(p)\\}\subset\mathbb{C}$ are
the Satake parameters of $\pi_{p}$. The standard $L$-function of $\pi$ is then

$L(s,\pi):=\prod_{p}L(s,\pi_{p})=\sum_{n\geq
1}\frac{\lambda_{\pi}(n)}{n^{s}},$

which converges absolutely when $\text{Re}(s)>1$. The sequence of coefficients
$\lambda_{\pi}(n)$ thus defined is multiplicative, with the property that

$\lambda_{\pi}(p^{r})=\sum_{\begin{subarray}{c}r_{1},\ldots,r_{m}\geq 0\\\
r_{1}+\cdots+r_{m}=r\end{subarray}}\prod_{1\leq j\leq
m}\alpha_{j,\pi}(p)^{r_{j}}.$

The generalized Ramanujan conjecture (GRC) implies that for all $1\leq j\leq
m$, $|\alpha_{j,\pi}(p)|=1$ whenever $p\nmid q_{\pi}$ and otherwise
$|\alpha_{j,\pi}(p)|\leq 1$. It follows that if $\pi$ satisfies GRC then

$|\lambda_{\pi}(p^{r})|\leq\sum_{\begin{subarray}{c}r_{1},\ldots,r_{m}\geq
0\\\ r_{1}+\cdots+r_{m}=r\end{subarray}}1=\binom{m+r-1}{r}=d_{m}(p^{r}),$

and therefore that $|\lambda_{\pi}(n)|\leq d_{m}(n)$. As a consequence of
these properties we will prove the following.

######  Theorem 1.2.

Let $m\geq 2$ and let $\pi$ be a fixed cuspidal automorphic representation for
$GL_{m}(\mathbb{A})$ as above. Assume that $\pi$ satisfies GRC. Let $10\leq
h_{0}\leq X/(10(\log X)^{m^{2}-1})$ and let $h:=h_{0}(\log X)^{m^{2}-1}$. Then
there is a constant $\theta=\theta(m)>0$ such that

$\frac{1}{X}\int_{X}^{2X}\left(\frac{1}{h}\sum_{x<n\leq
x+h}|\lambda_{\pi}(n)|^{2}-\frac{1}{X}\sum_{X<n\leq
2X}|\lambda_{\pi}(n)|^{2}\right)^{2}dx\ll\frac{\log\log h_{0}}{\log
h_{0}}+\frac{\log\log X}{(\log X)^{\theta}}.$

######  Remark 1.3.

In the case $m=2$ the parameter $h$ must grow faster than $(\log X)^{3}$ in
Theorem 1.2, whereas Corollary 1.1 allows any $h$ growing faster than $\log
X$. This is due to the fact that the range of $h$ in these estimates depends
on the size of $\sum_{p\leq X}|\lambda_{\pi}(p)|^{4}/p$. When $\pi=\pi_{f}$
for a cusp form $f$ on $GL_{2}(\mathbb{A})$ we may estimate this sum using the
well-known expression

$|\lambda_{f}(p)|^{4}=2+3\lambda_{\text{Sym}^{2}f}(p)+\lambda_{\text{Sym}^{4}f}(p)$

for all primes $p$, since $\text{Sym}^{r}f$ is cuspidal automorphic for
$r=2,4$ and thus $\sum_{p\leq X}\lambda_{\text{Sym}^{r}f}(p)/p=O(1)$. When
$m\geq 3$ such data for $|\lambda_{\pi}(p)|^{4}$ is not available
unconditionally in general, to the best of the author’s knowledge. Assuming
the validity of Langlands’ functoriality conjecture, a (likely more
complicated) expression would follow from the factorization of the standard
$L$-function $L(s,f)$ of the representation
$f=\pi\otimes\tilde{\pi}\otimes\pi\otimes\tilde{\pi}$, where $\tilde{\pi}$ is
the contragredient representation of $\pi$. Using GRC alone, we cheaply obtain
the simple upper bound

$\sum_{p\leq X}\frac{|\lambda_{\pi}(p)|^{4}}{p}\leq m^{2}\sum_{p\leq
X}\frac{|\lambda_{\pi}(p)|^{2}}{p}=m^{2}\log\log X+O(1),$

from Rankin-Selberg theory, and this is the source of the exponent $m^{2}$ in
the range of $h$.
We will instead deduce Corollary 1.1 from Theorem 1.4 below, which is tailored
to $GL_{2}$ cusp forms.

####  1.3.2. Moments of coefficients of $GL_{2}$ cusp forms in typical short
intervals

Our next application concerns short interval averages of the moments
$n\mapsto|\lambda_{f}(n)|^{\alpha}$, for any $\alpha>0$, with the notation of
the previous subsection. This generalizes Corollary 1.1.

######  Theorem 1.4.

Let $\alpha>0$ and define

$c_{\alpha}:=\frac{2^{\alpha}}{\sqrt{\pi}}\frac{\Gamma\left(\frac{\alpha+1}{2}\right)}{\Gamma(\alpha/2+2)},\quad
d_{\alpha}:=c_{2\alpha}-2c_{\alpha}+1.$

Let $10\leq h_{0}\leq X/(10(\log X)^{d_{\alpha}})$ and put $h:=h_{0}(\log
X)^{d_{\alpha}}$. There is a constant $\theta=\theta(\alpha)>0$ such that

$\frac{1}{X}\int_{X}^{2X}\left(\frac{1}{h}\sum_{x<n\leq
x+h}|\lambda_{f}(n)|^{\alpha}-\frac{1}{X}\sum_{X<n\leq
2X}|\lambda_{f}(n)|^{\alpha}\right)^{2}dx\ll_{\alpha}\left(\left(\frac{\log\log
h_{0}}{\log h_{0}}\right)^{c_{\alpha}}+\frac{\log\log X}{(\log
X)^{\theta}}\right)(\log X)^{2(c_{\alpha}-1)}.$

When $\alpha\neq 2$, the Rankin-Selberg theory is no longer available. In its
place, a crucial role in the proof of this result is played by a quantitative
version of the Sato-Tate theorem for non-CM cusp forms, due to Thorner [44],
which uses the deep results of Newton and Thorne [30]; see Section 5.1 for the
details.

######  Remark 1.5.

Using the Sato-Tate theorem and [23, Thm. 1.2.4] it can be shown that
$\frac{1}{X}\sum_{X<n\leq 2X}|\lambda_{f}(n)|^{\alpha}\gg_{\alpha}(\log
X)^{c_{\alpha}-1}$, so the estimate in Theorem 1.4 is indeed non-trivial.

####  1.3.3. Hooley’s $\Delta$-function in short intervals

The distribution of divisors of a typical positive integer is a topic of
classical interest, and a source of many difficult problems. Given an integer
$n\in\mathbb{N}$, let

$\mathcal{D}_{n}(v):=\frac{1}{d(n)}\sum_{\begin{subarray}{c}d|n\\\ d\leq
e^{v}\end{subarray}}1,\quad\quad\text{ for }v\in\mathbb{R}.$

This is a distribution function on the divisors of $n$. A concentration
function for $\mathcal{D}_{n}(v)$, in the sense of probability theory, can be
given by

$Q(n):=\max_{u\in\mathbb{R}}|\mathcal{D}_{n}(u+1)-\mathcal{D}_{n}(u)|=\max_{u\in\mathbb{R}}\frac{1}{d(n)}\sum_{\begin{subarray}{c}d|n\\\
e^{u}<d\leq e^{u+1}\end{subarray}}1.$

Hooley [15] considered the unnormalized variant

$\Delta(n):=d(n)Q(n)=\max_{u\in\mathbb{R}}\sum_{\begin{subarray}{c}d|n\\\
e^{u}<d\leq e^{u+1}\end{subarray}}1,$

now known as Hooley’s $\Delta$-function, and used it to attack various
problems related, among other things, to inhomogeneous Diophantine
approximation by squares, as well as Waring’s problem for cubes. Clearly,
$0\leq\Delta(n)\leq d(n)$, but one seeks more refined data about this
function. For example, Erdős [2] conjectured in 1948 that, except on a set of
natural density 0, $\Delta(n)>1$.
Many authors have investigated the average and almost sure behaviour of
$\Delta$. Maier and Tenenbaum [21] proved Erdős’ conjecture in a quantitative
form. A significant portion of Hall and Tenenbaum’s book [12] is devoted to
the $\Delta$ function, including the currently best known upper bound for its
mean value (see also [11]). For a partial survey of these results, see [40] .
Much less has been done concerning the local behaviour of the
$\Delta$-function. To the author’s knowledge the only result about its short
interval behaviour was worked out in the setting of polynomials over a finite
field by Gorodetsky [5, Cor. 1.5].
By relating $\Delta(n)$ to integral averages of the characteristic function of
$\mathcal{D}_{n}$ (which is multiplicative), we can deduce the following lower
bound for $\Delta$ on average over typical short intervals of length $(\log
X)^{1/2+\eta}$, for $\eta\in(0,1/2]$.

######  Corollary 1.6.

Fix $\delta\in(0,1]$, and let $10\leq h_{0}\leq\frac{X}{10(\log
X)^{(1+\delta)/2}}$ and set $h=h_{0}(\log X)^{(1+\delta)/2}$. Then for all but
$o_{h_{0}\rightarrow\infty}(X)$ integers $x\in[X/2,X]$ we have

$\frac{1}{h}\sum_{x-h<n\leq x}\Delta(n)\gg\delta\log\log X.$

###  1.4. Statement of main results

We fix $B,C\geq 1$, $0<A\leq B$, and for $X$ large we define
$\mathcal{M}(X;A,B,C)$ to denote the set of multiplicative functions
$f:\mathbb{N}\rightarrow\mathbb{C}$ such that:

  1. (i)

$|f(p)|\leq B$ for all primes $p\leq X$,

  2. (ii)

$|f(n)|\leq d_{B}(n)^{C}\text{ for all }n\leq X$,

  3. (iii)

for all $z_{0}\leq z\leq w\leq X$, we have

(6) $\sum_{z<p\leq w}\frac{|f(p)|}{p}\geq A\sum_{z<p\leq
w}\frac{1}{p}-O\left(\frac{1}{\log z}\right).$

As described above, the work [24] treats 1-bounded multiplicative functions
$f\in\mathcal{M}(X;A,1,1)$. We are interested in generalizing the results from
[24] to be applicable to the collection $\mathcal{M}(X;A,B,C)$, with $B\geq
1$. For the purpose of applications, we further extend $\mathcal{M}(X;A,B,C)$
as follows.

Fixing $\gamma>0$ and $0<\sigma\leq A$, we define
$\mathcal{M}(X;A,B,C;\gamma,\sigma)$ to be the collection of multiplicative
functions $f:\mathbb{N}\rightarrow\mathbb{C}$ satisfying (i) and (ii), as well
as the additional hypotheses

  * (iii’)

for all $z_{0}\leq z\leq w\leq X$, we have

(7) $\sum_{z<p\leq w}\frac{|f(p)|}{p}\geq A\sum_{z<p\leq
w}\frac{1}{p}-O\left(\frac{1}{(\log z)^{\gamma}}\right),$

  * (iv)

letting $t_{0}\in[-X,X]$ be a minimizer on $[-X,X]$ of the map

$t\mapsto\rho(f,n^{it};X)^{2}:=\sum_{p\leq
X}\frac{|f(p)|-\text{Re}(f(p)p^{-it})}{p},$

we have for all $t\in[-2X,2X]$ that

(8) $\rho(f,n^{it};X)^{2}\geq\sigma\min\\{\log\log X,\log(1+|t-t_{0}|\log
X)\\}-O_{A,B}(1).$

Condition (iii’) is a weaker form of (iii). The full strength of (iii) is
needed in [24, Lem. A.1] to obtain (iv) for all $0<\sigma<\sigma_{A}$ with a
particular constant $\sigma_{A}>0$, which is crucial to the proof of [24,
Theorem 1.9]. We will show below, as a consequence of [24, Lem. 5.1(i)], that
if $f\in\mathcal{M}(X;A,B,C)$ then condition (iv) here holds for any
$0<\sigma<\sigma_{A,B}$, where

(9) $\sigma_{A,B}:=\frac{A}{3}\left(1-\text{sinc}\left(\frac{\pi
A}{2B}\right)\right),\quad\quad\text{sinc}(t):=\frac{\sin t}{t}\text{ for
}t\neq 0.$

In particular, for any $0<\sigma<\sigma_{A,B}$,
$\mathcal{M}(X;A,B,C)\subseteq\mathcal{M}(X;A,B,C;1,\sigma)$. In proving
Corollary 1.1, for instance, it is profitable to assume (iii’) rather than
(iii), given currently available quantitative versions of the Sato-Tate
theorem (see (26) below).

In the sequel, fix $B,C\geq 1$, $0<A\leq B$, $\gamma>0$ and $0<\sigma\leq A$.
We define

(10)
$\hat{\sigma}:=\min\\{1,\sigma\\},\quad\quad\kappa:=\frac{\hat{\sigma}}{8B+21}.$

Given $T\geq 1$ we set

$M_{f}(X;T):=\min_{|t|\leq T}\rho(f,n^{it};X)^{2}=\min_{|t|\leq T}\sum_{p\leq
X}\frac{|f(p)|-\text{Re}(f(p)p^{-it})}{p}.$

We select $t_{0}(f,T)$ to be a real number $t\in[-T,T]$ that gives the minimum
in the definition of $M_{f}(X;T)$.
Finally, for a multiplicative function $f:\mathbb{N}\rightarrow\mathbb{C}$ we
recall that

$H(f;X):=\prod_{p\leq X}\left(1+\frac{(|f(p)|-1)^{2}}{p}\right),$

observing for future reference that whenever $|f(p)|\leq B$ for all $p\leq X$,

(11) $H(f;X)\asymp_{B}\prod_{p\leq
X}\left(1+\frac{|f(p)|^{2}-1}{p}\right)\left(1+\frac{|f(p)|-1}{p}\right)^{-2}.$

The main result of this paper is the following.

######  Theorem 1.7.

Let $X\geq 100$. Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$, and put
$t_{0}=t_{0}(f,X)$. Let $10\leq h_{0}\leq X/10H(f;X)$, and set
$h:=h_{0}H(f;X)$. Then

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f(n)-\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll_{A,B,C}\left(\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{A}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)^{2}.$

######  Corollary 1.8.

Let $X\geq 100$. Let $f\in\mathcal{M}(X;A,B,C)$ and put $t_{0}=t_{0}(f,X)$.
Let $10\leq h_{0}\leq X/10H(f;X)$, and set $h:=h_{0}H(f;X)$. Then

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f(n)-\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll_{A,B,C}\left(\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{A}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)^{2},$

for any $0<\kappa<\kappa_{A,B}:=\frac{\min\\{1,\sigma_{A,B}\\}}{16B+21}$.

######  Remark 1.9.

By Shiu’s theorem (Lemma 3.2 below), it is easy to show that the long sum term
in the LHS is

$\ll\frac{1}{X}\left(\sum_{X/3<n\leq X}|f(n)|\right)^{2}\ll_{B}\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)^{2}.$

Thus, this theorem shows that the variance is smaller than the square of the
“trivial” bound for the long sum by a factor tending to 0 provided
$h_{0}(X)\rightarrow\infty$, as $X\rightarrow\infty$.

##  2\. Outline of the proof of Theorem 1.7

To prove Theorem 1.7 we will establish two estimates. The first compares
typical short averages of $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$ to typical
medium-length ones (i.e., of length $X/(\log X)^{c}$, for $c=c(\sigma)>0$
small). The techniques involved were developed in [26], using certain
corresponding refinements that arose in [24].

######  Theorem 2.1.

Let $B,C\geq 1$, $0<A\leq B$, $\gamma>0$ and $0<\sigma\leq A$. Assume that
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Let $10\leq h_{0}\leq X/10H(f;X)$,
set $h_{1}:=h_{0}H(f;X)$ and $h_{2}=X/(\log X)^{\hat{\sigma}/2}$ and assume
that $h_{1}\leq h_{2}$. Finally, put $t_{0}=t_{0}(f,X)$. Then

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\sum_{x-h_{1}<m\leq
x}f(m)-\frac{1}{h_{1}}\int_{x-h_{1}}^{x}u^{it_{0}}du\cdot\frac{1}{h_{2}}\sum_{x-h_{2}<m\leq
x}f(m)m^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll_{A,B,C}\left(\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{A}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)^{2}.$

Essential to the treatment of Theorem 2.1 are strong pointwise upper bounds
for Dirichlet polynomials

$\sum_{X/3<n\leq X}\frac{a(n)f(n)}{n^{1+it}},$

where $\\{a(n)\\}_{n}\subset[0,1]$ is a particular sequence of weights,
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$ and $t\in[-X,X]$. To obtain these
estimates we will apply some of the recent results about unbounded
multiplicative functions described in Section 1.2. This is carried out at the
beginning of Section 3.

The second estimate we require towards Theorem 1.7 is a “Lipschitz” bound,
approximating the averages of a multiplicative function $f$ on any
sufficiently long medium-length interval by a long interval average of $f$.
The techniques involved are different from those used in the proof of Theorem
2.1, and largely follow the work of Granville, Harper and Soundararajan [7];
see Section 3.1 for the details.

######  Theorem 2.2.

Let $B,C\geq 1$, $0<A\leq B$, $\gamma>0$ and $0<\sigma\leq A$. Let $X/(\log
X)^{\hat{\sigma}/2}\leq h\leq X/10$, and let $x\in[X/2,X]$. Assume that
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Then the following bounds hold:

$\displaystyle\frac{1}{h}\sum_{x-h<n\leq x}f(n)n^{-it_{0}}$
$\displaystyle=\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}+O_{A,B,C}\left(\frac{(\log\log X)^{\hat{\sigma}+1}}{(\log
X)^{\hat{\sigma}/2}}\prod_{p\leq X}\left(1+\frac{|f(p)|-1}{p}\right)\right),$
$\displaystyle\frac{1}{h}\sum_{x-h<n\leq x}f(n)$
$\displaystyle=\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}$ $\displaystyle+O_{A,B,C}\left(\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}/2}}\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)\right).$

###### Proof of Theorem 1.7 assuming Theorem 2.1 and Theorem 2.2.

Assume the hypotheses of Theorem 1.7, and set $h^{\prime}:=X/(\log
X)^{\hat{\sigma}/2}$. If $h>h^{\prime}$ then Theorem 1.7 follows immediately
from the second estimate in Theorem 2.2. Thus, we may assume that $h\leq
h^{\prime}$. Applying Cauchy-Schwarz, we get

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f(n)-\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f(n)-\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{1}{h^{\prime}}\sum_{x-h^{\prime}<n\leq
x}f(n)n^{-it_{0}}\right|^{2}dx$ $\displaystyle+\sup_{X/2<x\leq
X}\left|\frac{1}{h^{\prime}}\sum_{x-h^{\prime}<n\leq
x}f(n)n^{-it_{0}}-\frac{2}{X}\sum_{X/2<n\leq X}f(n)n^{-it_{0}}\right|^{2}$
$\displaystyle=:T_{1}+T_{2},$

upon trivially bounding $h^{-1}|\int_{x-h}^{x}u^{it_{0}}du|\leq 1$. By Theorem
2.1 and the first estimate of Theorem 2.2,

$\displaystyle T_{1}$ $\displaystyle\ll_{A,B,C}\left(\left(\frac{\log\log
h}{\log h}\right)^{A}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\prod_{p\leq
X}\left(1+\frac{|f(p)|-1}{p}\right)^{2}$ $\displaystyle T_{2}$
$\displaystyle\ll_{A,B,C}\frac{(\log\log X)^{\hat{\sigma}+1}}{(\log
X)^{\hat{\sigma}/2}}\prod_{p\leq X}\left(1+\frac{|f(p)|-1}{p}\right)^{2}.$

Combining these bounds proves the claim. ∎

##  3\. Averages of Divisor-Bounded Multiplicative Functions and the proof of
Theorem 2.2

In the sequel we will require control over various averages of multiplicative
functions $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. In this and the next
subsection, such bounds are derived/recorded.
First, we will require some general pointwise estimates for prime power values
of $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. In preparation, define

$P(s):=\sum_{\begin{subarray}{c}n\geq 1\\\ p^{k}||n\Rightarrow p^{k}\leq
X\end{subarray}}\frac{f(n)}{n^{s}}=\prod_{p\leq
X}\left(1+\sum_{\begin{subarray}{c}k\geq 1\\\ p^{k}\leq
X\end{subarray}}\frac{f(p^{k})}{p^{ks}}\right),\quad\text{Re}(s)>1.$

Wherever $P$ is non-zero we may also write the logarithmic derivative
Dirichlet series

$-\frac{P^{\prime}}{P}(s)=\sum_{n\geq 1}\frac{\Lambda_{f}(n)}{n^{s}}.$

######  Lemma 3.1.

Suppose $f:\mathbb{N}\rightarrow\mathbb{C}$ is multiplicative and satisfies
$|f(n)|\leq d_{B}(n)^{C}$ for all $n\leq X$.
a) For any prime power $p^{\nu}\leq X$ we have

$|f(p^{\nu})|\ll_{B,C}\begin{cases}\left(\frac{5}{4}\right)^{\nu}\left(1+\frac{\nu}{B-1}\right)^{(B-1)C},&\text{
if $B>1$,}\\\ 1&\text{ if $B=1$.}\end{cases}$

b) For any $\eta\in[0,1/2)$ we have

$\sum_{\begin{subarray}{c}p^{\nu}\leq X\\\ \nu\geq
2\end{subarray}}\frac{|f(p^{\nu})|}{p^{(1-\eta)\nu}}\ll_{\eta,B,C}1.$

c) $\Lambda_{f}(n)=0$ unless $n=p^{\nu}$ for some prime power $p^{\nu}$. In
particular, if $p^{\nu}\leq X$ we have $|\Lambda_{f}(p)|\leq B\log p$ when
$\nu=1$ and otherwise
$|\Lambda_{f}(p^{\nu})|\ll_{\varepsilon,B,C}p^{\varepsilon\nu}$.

###### Proof.

a) If $B=1$ then the claim is obvious since $d_{B}\equiv 1$. Thus, we may
assume that $B>1$. We may also assume that $\nu$ is large relative to $B,C$,
for otherwise the estimate is trivial for a suitably large implicit constant.
Given these assumptions, observe that by Stirling’s approximation,

$\displaystyle|f(p^{\nu})|$ $\displaystyle\leq
d_{B}(p^{\nu})^{C}=\binom{\nu+B-1}{\nu}^{C}\ll_{B,C}\left(\frac{\sqrt{2\pi(\nu+B-1)}}{2\pi\sqrt{\nu(B-1)}}\left(1+\frac{B-1}{\nu}\right)^{\nu}\cdot\left(1+\frac{\nu}{B-1}\right)^{B-1}\right)^{C}$
$\displaystyle\ll_{B,C}\left(\frac{5}{4}\right)^{\nu}\left(1+\frac{\nu}{B-1}\right)^{C(B-1)},$

provided that $\nu$ is large enough that
$\left(1+\frac{B-1}{\nu}\right)^{C}\leq\frac{5}{4}$. This proves a).
b) Let $\delta:=\frac{1}{2}-\eta$. For each $2\leq p\leq X$ we have
$p^{1/2}>5/4$, and thus by a),

$\sum_{\begin{subarray}{c}\nu\geq 2:\\\ p^{\nu}\leq
X\end{subarray}}\frac{|f(p^{\nu})|}{p^{(1-\eta)\nu}}\ll_{B,C}\sum_{\nu\geq
2}\left(1+\frac{\nu}{B-1}\right)^{BC}\left(\frac{5}{4p^{1/2+\delta}}\right)^{\nu}\ll_{B,C,\delta}\sum_{\nu\geq
2}\left(\frac{5}{4p^{(1+\delta)/2}}\right)^{\nu}\ll p^{-1-\delta}.$

We deduce b) upon summing over $p\leq X$.
c) We begin by giving an expression for $\Lambda_{f}(p^{\nu})$.
In light of a), we may deduce that there is $\sigma=\sigma(B,C)>1$ such that
when $\text{Re}(s)\geq\sigma$,

(12) $\left|\sum_{p^{\nu}\leq X}\frac{f(p^{\nu})}{p^{\nu
s}}\right|\leq\frac{1}{2}\text{ for all $2\leq p\leq X$.}$

It follows from the Euler product representation of $P(s)$ that $P(s)\neq 0$
in the half-plane $\text{Re}(s)\geq\sigma$. Thus, $-P^{\prime}(s)/P(s)$ is
also analytic in this half-plane.
Integrating $-P^{\prime}/P$ term-by-term from $s$ to $\infty$ along a line
contained in the half-plane $\text{Re}(s)\geq\sigma$, we deduce that

$\sum_{n\geq 1}\frac{\Lambda_{f}(n)}{n^{s}\log n}=\log P(s)=\sum_{p\leq
X}\log\left(1+\sum_{\begin{subarray}{c}\nu\geq 1\\\ p^{\nu}\leq
X\end{subarray}}\frac{f(p^{\nu})}{p^{\nu s}}\right).$

Given (12), we obtain the Taylor expansion

$\displaystyle\sum_{n\geq 1}\frac{\Lambda_{f}(n)}{n^{s}\log n}$
$\displaystyle=\sum_{p\leq X}\sum_{k\geq
1}\frac{(-1)^{k-1}}{k}\sum_{\begin{subarray}{c}\nu_{1},\ldots,\nu_{k}\geq 1\\\
p^{\nu_{i}}\leq X\forall i\end{subarray}}\frac{f(p^{\nu_{1}})\cdots
f(p^{\nu_{k}})}{p^{(\nu_{1}+\cdots+\nu_{k})s}}$
$\displaystyle=\sum_{\begin{subarray}{c}p^{\nu}\\\ p\leq
X\end{subarray}}\frac{1}{p^{\nu s}\log p^{\nu}}\left(\log
p^{\nu}\cdot\sum_{1\leq
k\leq\nu}\frac{(-1)^{k-1}}{k}\sum_{\begin{subarray}{c}\nu_{1}+\cdots+\nu_{k}=\nu\\\
\nu_{1},\ldots,\nu_{k}\geq 1\\\ p^{\nu_{i}}\leq X\forall
i\end{subarray}}\prod_{1\leq i\leq k}f(p^{\nu_{i}})\right).$

By the identity theorem for Dirichlet series, we thus find that
$\Lambda_{f}(n)=0$ unless $n=p^{\nu}$ for some prime power $p^{\nu}$ with
$p\leq X$, in which case

(13) $\Lambda_{f}(p^{\nu})=\log p^{\nu}\cdot\sum_{1\leq
k\leq\nu}\frac{(-1)^{k-1}}{k}\sum_{\begin{subarray}{c}\nu_{1}+\cdot+\nu_{k}=\nu\\\
\nu_{1},\ldots,\nu_{k}\geq 1\\\ p^{\nu_{i}}\leq X\forall
i\end{subarray}}\prod_{1\leq i\leq k}f(p^{\nu_{i}}).$

When $\nu=1$ we get the expression $\Lambda_{f}(p)=f(p)\log p$, so that
$|\Lambda_{f}(p)|\leq B\log p$.
For $\nu\geq 2$ we simply note using the uniform bound
$d_{B}(n)^{C}\ll_{B,C,\varepsilon}n^{\varepsilon}$ and the triangle inequality
in (13) that

$|\Lambda_{f}(p^{\nu})|\ll_{B,C,\varepsilon}\sum_{1\leq
k\leq\nu}\frac{1}{k}\sum_{\begin{subarray}{c}\nu_{1}+\cdots+\nu_{k}=\nu\\\
\nu_{1},\ldots,\nu_{k}\geq 1\end{subarray}}\prod_{1\leq i\leq
k}p^{\nu_{i}\varepsilon}\leq p^{\nu\varepsilon}\mathfrak{p}(\nu),$

where $\mathfrak{p}(\nu)$ denotes the number of partitions of the positive
integer $\nu$. By a classical bound of Hardy-Ramanujan [13, Sec. 2.3], there
is an absolute constant $c>0$ such that

$\mathfrak{p}(\nu)\ll e^{c\sqrt{\nu}}\ll_{\varepsilon}p^{\nu\varepsilon},$

which implies the claim. ∎

We will use the following upper bound for non-negative functions repeatedly in
the sequel.

######  Lemma 3.2 (P. Shiu; [33], Thm. 1).

Let $f:\mathbb{N}\rightarrow\mathbb{C}$ be a multiplicative function
satisfying $|f(n)|\leq d_{B}(n)^{C}$ for all $n\leq X$. Let $\sqrt{X}<Y\leq
X$, $\delta\in(0,1)$ and let $Y^{\delta}\leq y\leq Y$. Then

$\sum_{Y-y<n\leq Y}|f(n)|\ll_{B,C,\delta}y\mathcal{P}_{f}(X).$

###### Proof.

The hypotheses required to apply [33, Thm. 1] are more precisely that
$|f(n)|\ll_{\varepsilon}n^{\varepsilon}$ for all $n\leq X$, and that there is
a constant $A\geq 1$ such that $|f(p^{\nu})|\leq A^{\nu}$ for all $p^{\nu}\leq
X$. The first hypothesis is obvious from $d_{B}(n)^{C}\leq d(n)^{\lceil
B\rceil C}\ll_{B,C,\varepsilon}n^{\varepsilon}$, while the second is implied
by Lemma 3.1 a). ∎

######  Lemma 3.3.

Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$ and let $t_{0}=t_{0}(f,X)$. Let
$X^{1/5}\leq Y\leq X$, and let $2\leq P\leq Q\leq\exp\left(\frac{\log
X}{\log\log X}\right)$. Then for any $1\leq Z\leq\log X$,

$\sup_{Z<|u|\leq X/2}\left|\sum_{\begin{subarray}{c}n\leq Y\\\ p|n\Rightarrow
p\notin[P,Q]\end{subarray}}f(n)n^{-i(t_{0}+u)}\right|\ll_{A,B,C}Y\mathcal{P}_{f}(X)\left(\left(\frac{\log
Q}{\log P}\right)^{2B}\frac{\log\log X}{\log
X^{\sigma}}+\frac{1}{\sqrt{Z}}\right).$

###### Proof.

Define $\beta(n):=f(n)1_{p|n\Rightarrow p\notin[P,Q]}$, and for
$t\in\mathbb{R}$ set $\beta_{t}(n):=\beta(n)n^{-it}$. Note that
$|\beta_{t}(n)|\leq|f(n)|$ for all $n$ and $t\in\mathbb{R}$. As
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$, we have that

  1. (1)

$\max_{p\leq X}|f(p)|\leq B$

  2. (2)

$\sum_{\begin{subarray}{c}p^{k}\leq X\\\ k\geq
2\end{subarray}}\frac{|f(p^{k})|\log p^{k}}{p^{k}}\ll_{B,C}1$ by Lemma 3.1b),
and

  3. (3)

$\sum_{y<p\leq X}\frac{|f(p)|}{p}\geq A\log\left(\frac{\log X}{\log
y}\right)-O_{A,B}(1)$ uniformly in $2\leq y\leq X$,

for all $t\in\mathbb{R}$. Thus, the hypotheses of [42, Cor. 2.1] are fulfilled
with $r=|f|$. Applying that result gives, for every $Z<|u|\leq X/2$,

$\left|\sum_{n\leq Y}\beta_{t_{0}+u}(n)\right|\ll_{A,B,C}\left(\sum_{n\leq
Y}|f(n)|\right)\left((1+M_{\beta_{t_{0}+u}(Y;Z/2)}e^{-M_{\beta_{t_{0}+u}}(Y;Z/2)}+\frac{1}{\sqrt{Z}}\right).$

Let $t=t(u)\in[-Z/2,Z/2]$ be the minimizer implicit in
$M_{f_{t_{0}+u}}(Y;Z/2)$, so that

$M_{\beta_{t_{0}+u}}(Y;Z/2)=\rho(\beta,n^{i(t_{0}+u+t)};Y)^{2}\leq 2B\log\log
X.$

As $X^{1/5}\leq Y\leq X$,

$\rho(\beta,n^{i(t_{0}+u+t)};Y)^{2}=\rho(\beta,n^{i(t_{0}+u+t)};X)^{2}-O_{B}(1)\geq\rho(f,n^{i(t_{0}+u+t)};X)^{2}-2B\log\left(\frac{\log
Q}{\log P}\right)-O_{B}(1).$

Since $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$ and $|t_{0}+u+t|\leq 2X$ and
$|u+t|>Z/2$, we have by (8) that

$\rho(f,n^{i(t_{0}+u+t)};X)^{2}\geq\sigma\min\\{\log\log X,\log(1+|u+t|\log
X)\\}-O_{A,B}(1)\geq\sigma\log\log X-O_{A,B}(1).$

It thus follows that

$\max_{Z<|u|\leq X/2}\left|\sum_{n\leq
Y}\beta_{t_{0}+u}(n)\right|\ll_{A,B,C}\left(\sum_{n\leq
Y}|f(n)|\right)\left(\left(\frac{\log Q}{\log P}\right)^{2B}\frac{\log\log
X}{(\log X)^{\sigma}}+\frac{1}{\sqrt{Z}}\right).$

Finally, applying Lemma 3.2 together with Mertens’ theorem, we obtain

$\sum_{n\leq Y}|f(n)|\ll_{B,C}Y\prod_{p\leq
Y}\left(1+\frac{|f(p)|-1}{p}\right)\ll_{B}Y\mathcal{P}_{f}(X),$

and the claim follows. ∎

We need the following estimate for certain divisor-bounded functions on
$y$-smooth666By a _$y$ -smooth_ or _$y$ -friable_ integer we mean a positive
integer $n$ such that $p|n\Rightarrow p\leq y$. integers, which is essentially
due to Tenenbaum and Wu [43, Cor. 2.2], improving on work of Song [35]. An
important rôle is played by the function $\rho_{k}(u)$ for $k\in\mathbb{N}$
and $u\geq 0$, which is a generalization of the classical Dickman-de Bruijn
function, defined by the differential delay equation

$u\rho_{k}^{\prime}(u)=(k-1)\rho_{k}(u)-k\rho_{k}(u-1)\text{ if }u\geq 1,$

and $\rho_{k}(u):=u^{k-1}/\Gamma(k)$ for $0\leq u<1$.

######  Lemma 3.4.

Let $g:\mathbb{N}\rightarrow\mathbb{R}$ be a non-negative multiplicative
function for which there are real constants $\delta\in(0,1)$, $\eta\in(0,1/2)$
and $D>0$, and an integer $k\geq 1$ such that

(14) $\displaystyle\sum_{p\leq z}g(p)\log p=kz+O(z/(\log z)^{\delta})\text{
for all }z\geq 2,$ (15) $\displaystyle\sum_{p^{\nu},\nu\geq
2}\frac{g(p^{\nu})}{p^{(1-\eta)\nu}}\leq D.$

Let $x\geq 3$ and let $\exp\left((\log x\log\log x)^{2/(2+\delta)}\right)\leq
y\leq x$. Set $u:=\frac{\log x}{\log y}$. Then

$\sum_{\begin{subarray}{c}n\leq x\\\ P^{+}(n)\leq
y\end{subarray}}g(n)=e^{-\gamma k}x\rho_{k}(u)\frac{G(1,y)}{\log
y}\left(1+O\left(\frac{\log(u+1)}{(\log y)^{\delta/2}}\right)\right),$

where $G(s,y):=\sum_{P^{+}(n)\leq y}g(n)n^{-s}$ for $\text{Re}(s)>0$, and
$P^{+}(n)$ denotes the largest prime factor of $n$.
In particular, for $y$ in the given range and such that $u\rightarrow\infty$
we have

$\sum_{\begin{subarray}{c}n\leq x\\\ P^{+}(n)\leq
y\end{subarray}}g(n)\ll_{k,D,\delta}x(\log y)^{k-1}\exp\left(-\frac{1}{3}u\log
u\right).$

###### Proof.

The first claim is a special case of [43, Cor. 2.2].
For the second claim, we note that

$G(1,y)\leq\exp\left(\sum_{p\leq
y}\frac{g(p)}{p}+\sum_{\begin{subarray}{c}p\leq y\\\ \nu\geq
2\end{subarray}}\frac{g(p^{\nu})}{p^{\nu}}\right)\ll_{D}\exp\left(\sum_{p\leq
y}\frac{g(p)}{p}\right)\ll_{\delta}(\log y)^{k},$

where the penultimate estimate follows from (15), and the last estimate is
obtained by partial summation from (14). Furthermore, by [34, (3.10)] and
well-known upper bounds for the Dickman-de Bruijn function (e.g., [6, (1.6)]),
we have

$\rho_{k}(u)=k^{u+O(u/\log(1+u))}\rho(u)\leq\exp\left(2u\log
k-\frac{1}{2}u\log u\right)\leq\exp\left(-\frac{1}{3}u\log u\right),$

whenever $u$ is large enough in terms of $k$, and the claim follows. ∎

By combining the last two lemmas, we may deduce the following upper bound for
Dirichlet polynomials of a special type (cf. [26, Lem. 3]).

######  Corollary 3.5.

Let $10\leq P\leq Q\leq\exp\left(\frac{\log X}{\log\log X}\right)$, and let
$1\leq Z\leq\log X$. Assume the hypotheses of Lemma 3.3, and assume $X\geq
X_{0}(B,C)$. Then for any $\sqrt{X}\leq Y\leq X$,

$\displaystyle\sup_{Z<|u|\leq X/2}\left|\sum_{Y/3<n\leq
Y}\frac{f(n)}{n^{1+i(t_{0}+u)}(1+\omega_{[P,Q]}(n))}\right|$
$\displaystyle\ll_{A,B,C}\mathcal{P}_{f}(X)\left(\left(\frac{\log Q}{\log
P}\right)^{3B}\frac{\log\log X}{(\log X)^{\sigma}}+\left(\frac{\log Q}{\log
P}\right)^{B}\frac{1}{\sqrt{Z}}\right),$

where $\omega_{[P,Q]}(n):=\sum_{\begin{subarray}{c}p|n\\\ P\leq p\leq
Q\end{subarray}}1$.

###### Proof.

Fix $u\in[-Z,Z]$ and set $t:=t_{0}+u$. Write $f=\alpha\ast\beta$, where
$\alpha$ and $\beta$ are multiplicative functions with
$\alpha(p^{k})=f(p^{k})$ whenever $P\leq p\leq Q$ and $p^{k}\leq X$, and
$\beta(p^{k})=f(p^{k})$ for all other primes powers $p^{k}\leq X$. We apply
the hyperbola method with $M=\sqrt{Y}$ to get

$\displaystyle\left|\sum_{Y/3<n\leq
Y}\frac{f(n)}{n^{1+it}(1+\omega_{[P,Q]}(n))}\right|$
$\displaystyle\ll\sum_{a\leq M}\frac{|\alpha(a)|}{a}\left|\sum_{Y/(3a)<b\leq
Y/a}\frac{\beta(b)b^{-it}}{b}\right|+\sum_{b\leq
Y/M}\frac{1}{b}\sum_{\max\\{M,Y/(3b)\\}<a\leq Y/b}\frac{|\alpha(a)|}{a}$
$\displaystyle=:\mathcal{R}_{1}+\mathcal{R}_{2}.$

To bound $\mathcal{R}_{1}$ we apply partial summation and Lemma 3.3 to obtain,
uniformly in $u$,

$\displaystyle\sum_{Y/(3a)<b\leq Y/a}\frac{\beta(b)b^{-it}}{b}$
$\displaystyle\ll_{A,B,C}\sup_{Y/(3a)\leq y\leq
Y/a}\frac{1}{y}\left|\sum_{n\leq y}\beta(b)b^{-it}\right|$
$\displaystyle\ll\mathcal{P}_{f}(X)\left(\left(\frac{\log Q}{\log
P}\right)^{2B}\frac{\log\log X}{(\log X)^{\sigma}}+\frac{1}{\sqrt{Z}}\right).$

Given the prime power support of $\alpha$, we have

$\sum_{a\leq M}\frac{|\alpha(a)|}{a}\ll_{B,C}\prod_{P\leq p\leq
Q}\left(1+\frac{|f(p)|}{p}\right)\ll_{B}\left(\frac{\log Q}{\log
P}\right)^{B},$

so that on combining this with the previous estimate, we obtain

$\mathcal{R}_{1}\ll_{A,B,C}\mathcal{P}_{f}(X)\left(\left(\frac{\log Q}{\log
P}\right)^{3B}\frac{\log\log X}{(\log X)^{\sigma}}+\left(\frac{\log Q}{\log
P}\right)^{B}\frac{1}{\sqrt{Z}}\right).$

Next, consider $\mathcal{R}_{2}$. Note that $\alpha(n)=f(n)1_{p|n\Rightarrow
p\in[P,Q]}$, and so since $|f(n)|\leq d_{B}(n)^{C}$ uniformly over $n\leq X$
we have

$\sum_{X/(3b)<a\leq
X/b}\frac{|\alpha(a)|}{a}\ll\frac{b}{X}\sum_{\begin{subarray}{c}n\leq X/b\\\
P^{+}(n)\leq Q\end{subarray}}g(n),$

where $g(n):=d_{\lceil B\rceil}(n)^{\lceil C\rceil}$. Note that $g(n)$ takes
integer values, and in particular taking $k:=\lceil B\rceil^{\lceil
C\rceil}\in\mathbb{Z}$, we have

$\sum_{p\leq X}g(p)\log p=k\sum_{p\leq X}\log p=kX+O(X/(\log X)^{1/2}),$

say, by the prime number theorem. Furthermore, that $g$ satisfies (15) with
some $\eta\in(0,1/2)$ and $D=O_{B,C}(1)$ is the content of Lemma 3.1b). Hence,
as $u_{b}:=\log(X/b)/\log Q\geq\frac{\log X}{2\log Q}\geq\frac{1}{2}\log\log
X$, Lemma 3.4 implies that when $X$ is sufficiently large in terms of $B$ and
$C$ we obtain

$\frac{b}{X}\sum_{\begin{subarray}{c}n\leq X/b\\\ P^{+}(n)\leq
Q\end{subarray}}g(n)\ll(\log Q)^{k-1}\exp\left(-\frac{1}{6}u_{b}\log
u_{b}\right)\ll_{B,C}(\log X)^{-200}.$

Combining this with the bound

$\sum_{b\leq X/M}\frac{|\beta(b)|}{b}\leq\sum_{b\leq
X/M}\frac{|f(b)|}{b}\ll(\log X)\mathcal{P}_{f}(X),$

which again follows from partial summation and Lemma 3.2, we obtain
$\mathcal{R}_{2}\ll(\log X)^{-100}$. Altogether, we conclude that

$\left|\sum_{Y/3<n\leq
Y}\frac{f(n)}{n^{1+i(t_{0}+u)}(1+\omega_{[P,Q]}(n))}\right|\ll_{B,C}\mathcal{P}_{f}(X)\left((\log
X)^{-100}+\left(\frac{\log Q}{\log P}\right)^{3B}\frac{\log\log X}{(\log
X)^{\sigma}}+\left(\frac{\log Q}{\log P}\right)^{B}\frac{1}{\sqrt{Z}}\right),$

uniformly over $Z<|u|\leq X/2$, and the claim follows since $Z^{-1/2}\geq(\log
X)^{-1/2}$. ∎

###  3.1. Lipschitz Bounds and Main Terms

In this subsection we derive a slight refinement of the Lipschitz bounds found
in [7, Thm. 1.5]. Specifically, our estimates are sensitive to the
distribution of values $|f(p)|$, which will allow us to obtain Theorem 2.2.
See also [28] for some related Lipschitz-type bounds for unbounded
multiplicative functions that share some overlap with the general result
obtained presently.

######  Theorem 3.6 (Relative Lipschitz bounds).

Let $1\leq w\leq X^{1/3}$ and let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$.
Set $t_{0}=t_{0}(f,X)$. Then

$\displaystyle\left|\frac{w}{X}\sum_{n\leq
X/w}f(n)n^{-it_{0}}-\frac{1}{X}\sum_{n\leq
X}f(n)n^{-it_{0}}\right|\ll_{A,B,C}\log\left(\frac{\log
X}{\log(ew)}\right)\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\hat{\sigma}}\mathcal{P}_{f}(X),$

where $\hat{\sigma}:=\min\\{1,\sigma\\}$. The same bound holds for the
quantity

$\left|\left(\frac{w}{X}\right)^{1+it_{0}}\sum_{n\leq
X/w}f(n)-\frac{1}{X^{1+it_{0}}}\sum_{n\leq X}f(n)\right|.$

To this end, we need to introduce some notation that is consistent with the
notation from [7]. For $\text{Re}(s)>1$ we write

$\mathcal{F}(s)=\sum_{\begin{subarray}{c}n\geq 1\\\ p^{k}||n\Rightarrow
p^{k}\leq X\end{subarray}}\frac{f(n)n^{-it_{0}}}{n^{s}}.$

For each prime power $p^{k}\leq X$, $k\geq 1$, define

$s(p^{k}):=\begin{cases}f(p^{k})p^{-ikt_{0}}&\text{ if $p\leq y$}\\\ 0&\text{
if $p>y$}\end{cases}\quad\quad\ell(p^{k}):=\begin{cases}0&\text{ if $p\leq
y$}\\\ f(p^{k})p^{-ikt_{0}}&\text{ if $p>y$}\end{cases},$

where $y\geq 2$ is a large parameter to be chosen later. We extend $s$ and
$\ell$ multiplicatively to all $n\in\mathbb{N}$ with $p^{k}||n\Rightarrow
p^{k}\leq X$, and set $s(n)=\ell(n)=0$ otherwise. For $\text{Re}(s)>1$, also
define

$\mathcal{S}(s)=\sum_{n\geq
1}\frac{s(n)}{n^{s}},\quad\quad\mathcal{L}(s):=\sum_{n\geq
1}\frac{\ell(n)}{n^{s}}.$

We recall that $\Lambda_{\ell}(n)$ is $n$th coefficient of the Dirichlet
series $-\mathcal{L}^{\prime}/\mathcal{L}(s)$, the logarithmic derivative of
$\mathcal{L}(s)$, which is well-defined for all $\text{Re}(s)>1$ whenever
$y\geq y_{0}(B,C)$ by Lemma 3.1 c).

######  Lemma 3.7.

Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Let $t\in\mathbb{R}$ and
$\xi\geq 1/\log X$. Then

$|\mathcal{F}(1+\xi+it)|\ll_{A,B,C}\xi^{-1}(1+\xi\log
X)^{1-A}\mathcal{P}_{f}(X)e^{-\rho(fn^{-it_{0}},n^{it};e^{1/\xi})^{2}}\ll_{B}(\log
X)^{B}.$

###### Proof.

By Lemma 3.1b) and $\max_{p\leq X}|f(p)|\leq B$, we deduce that

$\displaystyle|\mathcal{F}(1+\xi+it)|$ $\displaystyle\ll_{B}\prod_{p\leq
X}\left|1+\frac{f(p)p^{-i(t_{0}+t)}}{p^{1+\xi}}\right|\exp\left(\sum_{\begin{subarray}{c}p^{k}\leq
X\\\ k\geq 2\end{subarray}}\frac{|f(p^{k})|}{p^{k(1+\xi)}}\right)$
$\displaystyle\ll_{B,C}\prod_{p\leq
X}\left(1+\frac{|f(p)|}{p^{1+\xi}}\right)\left(1+\frac{\text{Re}(f(p)p^{-i(t_{0}+t)})-|f(p)|}{p^{1+\xi}}\right).$

By partial summation and the prime number theorem, the estimates

$\displaystyle\sum_{p>e^{1/\xi}}\frac{\alpha_{p}}{p^{1+\xi}}$
$\displaystyle\ll_{B}\int_{e^{1/\xi}}^{\infty}e^{-\xi v}\frac{dv}{v}\ll_{B}1$
(16) $\displaystyle\sum_{p\leq
e^{1/\xi}}\left(\frac{\alpha_{p}}{p}-\frac{\alpha_{p}}{p^{1\pm\xi}}\right)$
$\displaystyle\ll B\xi\sum_{p\leq e^{1/\xi}}\frac{\log p}{p}\ll_{B}1,$

are valid for any sequence $\\{\alpha_{p}\\}_{p}\subset\mathbb{C}$ with
$\max_{p}|\alpha_{p}|\ll_{B}1$. It follows that

$\displaystyle|\mathcal{F}(1+\xi+it)|$ $\displaystyle\ll_{B,C}\prod_{p\leq
e^{1/\xi}}\left(1+\frac{|f(p)|}{p}\right)\exp\left(-\sum_{p\leq
e^{1/\xi}}\frac{|f(p)|-\text{Re}(f(p)p^{-i(t_{0}+t)})}{p}\right)$
$\displaystyle\ll_{B}\xi^{-1}\mathcal{P}_{f}(X)\exp\left(\sum_{e^{1/\xi}<p\leq
X}\frac{1-|f(p)|}{p}\right)e^{-\rho(fn^{-it_{0}},n^{it};e^{1/\xi})^{2}}.$

Since $f\in\mathcal{M}(X;A,B;\gamma,\sigma)$, we have

$\sum_{e^{1/\xi}<p\leq X}\frac{1-|f(p)|}{p}\leq(1-A)\sum_{e^{1/\xi}<p\leq
X}\frac{1}{p}+O(\xi^{\gamma})=(1-A)\log(\xi\log X)+O_{A}(1).$

We thus obtain

$|\mathcal{F}(1+\xi+it)|\ll_{A,B,C}\xi^{-1}(1+\xi\log
X)^{1-A}\mathcal{P}_{f}(X)e^{-\rho(fn^{-it_{0}},n^{it};e^{1/\xi})^{2}},$

which proves the first claimed estimate.
To obtain the second, note that $\rho(fn^{-it_{0}},n^{it};Y)^{2}\geq 0$ for
all $Y\geq 2$, and so using $|f(p)|\leq B$ and $A\geq 0$ we obtain the further
bound

$\displaystyle\ll\xi^{-1}(\xi\log X)^{1-A}\mathcal{P}_{f}(X)\ll(\log
X)\mathcal{P}_{f}(X)$ $\displaystyle\ll_{B}\exp\left(\sum_{p\leq
X}\frac{|f(p)|}{p}\right)\ll(\log X)^{B},$

as claimed. ∎

To bound certain error terms in the proof of Theorem 3.6 we require the
following estimate, whose proof largely follows that of [7, Lem. 2.4]

######  Lemma 3.8.

Let $1\leq w\leq X^{1/3}$, $w\leq y\leq\sqrt{X}$ and $\eta:=1/\log y$. Then
for any $X/w\leq Z\leq X$,

$\sum_{mn\leq Z}|s(m)|\frac{|\ell(n)|}{n^{\eta}}+\int_{0}^{\eta}\sum_{mkn\leq
Z}|s(m)|\frac{|\Lambda_{\ell}(k)||\ell(n)|}{k^{\alpha}n^{2\eta+\alpha}}d\alpha\ll_{A,B,C}Z\left(\frac{\log
y}{\log Z}\right)^{A}\mathcal{P}_{f}(X).$

###### Proof.

In the first sum the summands arise from a Dirichlet convolution of
multiplicative functions $|s|\ast|\ell|n^{-\eta}$, and we clearly have
$|s(n)|,|\ell(n)|\leq d_{B}(n)^{C}$ for all $n\leq X$. By Lemma 3.2 and (3.1),
the first sum is therefore

$\ll_{B,C}\frac{Z}{\log Z}\exp\left(\sum_{p\leq
y}\frac{|f(p)|}{p}+\sum_{y<p\leq
Z}\frac{|f(p)|}{p^{1+\eta}}\right)\ll_{B}Z\frac{\log y}{\log
X}\mathcal{P}_{f}(X)\exp\left(\sum_{y<p\leq X}\frac{1-|f(p)|}{p}\right).$

Arguing as in the previous lemma, since
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$ this is bounded by

$\ll_{A}Z\frac{\log y}{\log X}\cdot\left(\frac{\log X}{\log
y}\right)^{1-A}\mathcal{P}_{f}(X)\ll Z\left(\frac{\log y}{\log
X}\right)^{A}\mathcal{P}_{f}(X).$

For the second term, we use Lemma 3.1c), which shows
that$|\Lambda_{\ell}(p)|\leq B\log p$ and
$|\Lambda_{\ell}(p^{\nu})|\ll_{\varepsilon,B,C}p^{\nu\varepsilon}$. It follows
that for $1\leq K\leq X$,

$\left|\sum_{k\leq K}\Lambda_{\ell}(n)n^{-\alpha}\right|\leq B\sum_{p\leq
K}p^{-\alpha}\log p+O_{B,C}\left(\sum_{\begin{subarray}{c}p^{\nu}\leq K\\\
\nu\geq 2\end{subarray}}p^{\nu/6}\right)\ll_{B,C}K^{1-\alpha}+K^{2/3},$

say, which is acceptable. Therefore taking $K=Z/mn$, the $\alpha$ integral in
the statement is

$\ll_{B,C}\int_{0}^{\eta}Z^{1-\alpha}\sum_{mn\leq
Z}\frac{|s(m)|}{m^{1-\alpha}}\frac{|\ell(n)|}{n^{1+2\eta}}d\alpha.$

Extending the inner sum by positivity to all products $mn$ with
$p^{k}||mn\Rightarrow p^{k}\leq Z$ and using the estimates (3.1) once again,
we may bound the integral using Euler products as

$\displaystyle\ll_{B,C}\int_{0}^{\eta}Z^{1-\alpha}\cdot\prod_{p\leq
y}\left(1+\frac{|f(p)|}{p^{1-\alpha}}\right)\prod_{y<p\leq
X}\left(1+\frac{|f(p)|}{p^{1+2\eta}}\right)d\alpha$
$\displaystyle\ll_{B}Z\prod_{p\leq
y}\left(1+\frac{|f(p)|}{p}\right)\int_{0}^{\eta}Z^{-\alpha}d\alpha\ll_{B}Z\frac{\log
y}{\log Z}\mathcal{P}_{f}(X)\exp\left(\sum_{y<p\leq
X}\frac{1-|f(p)|}{p}\right)$ $\displaystyle\ll_{A,B}Z\left(\frac{\log y}{\log
X}\right)^{A}\mathcal{P}_{f}(X).$

This completes the proof. ∎

###### Proof of Theorem 3.6.

The proof follows that of [7, Thm. 1.5], and we principally highlight the
differences.
We begin with the first claim of the theorem. Let $T:=(\log X)^{B+1}$ and
$y:=\max\\{ew,T^{2}\\}$. Fix $\eta:=1/\log y$, $c_{0}:=1+1/\log X$. Then [7,
Lem. 2.2] (replacing $\beta$ by $\beta/2$) and Lemma 3.8 combine to show that

$\displaystyle\frac{1}{X}\sum_{n\leq X}f(n)n^{-it_{0}}-\frac{w}{X}\sum_{n\leq
X/w}f(n)n^{-it_{0}}$ $\displaystyle=\int_{0}^{\eta}\int_{0}^{\eta}\frac{1}{\pi
i}\int_{c_{0}-i\infty}^{c_{0}+i\infty}\mathcal{S}(s)\mathcal{L}(s+\alpha)\frac{\mathcal{L}^{\prime}}{\mathcal{L}}(s+\alpha)\frac{\mathcal{L}^{\prime}}{\mathcal{L}}(s+\alpha+2\beta)\frac{X^{s-1}(1-w^{1-s})}{s}dsd\beta
d\alpha$ $\displaystyle+O_{A,B,C}\left(\left(\frac{\log y}{\log
X}\right)^{A}\mathcal{P}_{f}(X)\right).$

Consider the inner integral over $s$. Shifting $s\mapsto s-\alpha-\beta$ and
applying [7, Lem. 2.5], this is

$\displaystyle\frac{1}{\pi
i}\int_{c_{0}-iT}^{c_{0}+iT}\mathcal{S}(s-\alpha-\beta)\mathcal{L}(s+\beta)\left(\sum_{y<m<X/y}\frac{\Lambda_{\ell}(m)}{m^{s-\beta}}\right)\left(\sum_{y<n<X/y}\frac{\Lambda_{\ell}(n)}{n^{s+\beta}}\right)$
$\displaystyle\frac{X^{s-1-\alpha-\beta}(1-w^{1+\alpha+\beta-s})}{s-\alpha-\beta}ds$
$\displaystyle+O\left(\frac{1}{\log X}\right).$

Extracting the maximum over $|t|\leq T$, then applying Cauchy-Schwarz and [7,
Lem. 2.6], the main term for the $s$-integral is bounded above by

$\displaystyle\ll X^{-\alpha-\beta}\left(\max_{|t|\leq
T}\frac{|\mathcal{S}(c_{0}-\alpha-\beta+it)\mathcal{L}(c_{0}+\beta+it)||1-w^{1+\alpha+\beta-
c_{0}-it}|}{|c_{0}-\alpha-\beta+it|}\right)$
$\displaystyle\cdot\left(\int_{-T}^{T}\left|\sum_{y<m<X/y}\frac{\Lambda_{\ell}(m)}{m^{c_{0}-\beta-
it}}\right|^{2}dt\right)^{1/2}\left(\int_{-T}^{T}\left|\sum_{y<m<X/y}\frac{\Lambda_{\ell}(m)}{m^{c_{0}+\beta-
it}}\right|^{2}dt\right)^{1/2}$
$\displaystyle\ll_{B,C}X^{-\alpha-\beta}\left(\max_{|t|\leq
T}\frac{|\mathcal{S}(c_{0}-\alpha-\beta+it)\mathcal{L}(c_{0}+\beta+it)||1-w^{1+\alpha+\beta-
c_{0}-it}|}{|c_{0}-\alpha-\beta+it|}\right)$
$\displaystyle\cdot\left(\sum_{y<p<X/y}\frac{\log
p}{p^{c_{0}-2\beta}}+y^{-1/2+2\beta}\right)^{1/2}\left(\sum_{y<p<X/y}\frac{\log
p}{p^{c_{0}+2\beta}}+y^{-1/2-2\beta}\right)^{1/2}$ $\displaystyle\ll
X^{-\alpha-\beta}\left(\frac{X}{y}\right)^{\beta}\min\\{\log
X,1/\beta\\}\left(\max_{|t|\leq
T}\frac{|\mathcal{S}(c_{0}-\alpha-\beta+it)\mathcal{L}(c_{0}+\beta+it)||1-w^{1+\alpha+\beta-
c_{0}-it}|}{|c_{0}-\alpha-\beta+it|}\right).$

Furthermore, by (3.1) and $\alpha,\beta\leq\eta=1/\log y$, we see that for any
$t\in\mathbb{R}$,

$\displaystyle
X^{-\alpha-\beta}\left(\frac{X}{y}\right)^{\beta}|\mathcal{S}(c_{0}-\alpha-\beta+it)\mathcal{L}(c_{0}+\beta+it)|$
$\displaystyle\ll_{B,C}X^{-\alpha}y^{-\beta}|\mathcal{S}(c_{0}+\beta+it)\mathcal{L}(c_{0}+\beta+it)|$
$\displaystyle\ll X^{-\alpha}|\mathcal{F}(c_{0}+\beta+it)|.$

Thus, we have so far shown that

$\displaystyle\left|\frac{1}{X}\sum_{n\leq
X}f(n)n^{-it_{0}}-\frac{w}{X}\sum_{n\leq X/w}f(n)n^{-it_{0}}\right|$ (17)
$\displaystyle\ll_{A,B,C}\int_{0}^{\eta}\int_{0}^{\eta}X^{-\alpha}\min\\{\log
X,1/\beta\\}\max_{|t|\leq
T}\frac{|\mathcal{F}(c_{0}+\beta+it)(1-w^{1+\alpha+\beta-
c_{0}-it})|}{|c_{0}+\beta+it|}d\beta d\alpha$
$\displaystyle+\left(\frac{\log\log X+\log(ew)}{\log
X}\right)^{A}\mathcal{P}_{f}(X)+\frac{1}{\log X}.$

Observe next that

$|w^{-\beta-it}-w^{1+\alpha+\beta-
c_{0}-it}|\ll\left(\alpha+\beta+\frac{1}{\log X}\right)\log(ew),$

so that we may rewrite the integral expression in (17) as

$\displaystyle\int_{0}^{\eta}\left(\int_{0}^{\eta}X^{-\alpha}dx\right)\min\\{\log
X,1/\beta\\}\max_{|t|\leq T}\frac{|\mathcal{F}(c_{0}+\beta+it)(1-w^{-\beta-
it})|}{|c_{0}+\beta+it|}d\beta$
$\displaystyle+(\log(ew))\int_{0}^{\eta}\max_{|t|\leq
T}|\mathcal{F}(c_{0}+\beta+it)|\min\\{\log
X,\beta^{-1}\\}\int_{0}^{\eta}X^{-\alpha}\left(\alpha+\beta+1/\log
X\right)d\alpha d\beta$ $\displaystyle=:T_{1}+T_{2}.$

We first estimate $T_{2}$. The integral over $\alpha$ is

$\ll\left(\beta+\frac{1}{\log
X}\right)\int_{0}^{\eta}X^{-\alpha}d\alpha+\int_{0}^{\eta}\alpha
X^{-\alpha}d\alpha\ll\frac{1}{\log X}\left(\beta+\frac{1}{\log X}\right).$

Applying Lemma 3.7 (with $\xi=1/\log X+\beta$) and
$\rho(fn^{-it_{0}},n^{it};Y)^{2}\geq 0$ for all $Y\geq 2$,

$\displaystyle T_{2}$
$\displaystyle\ll_{A,B,C}(\log(ew))\mathcal{P}_{f}(X)\int_{0}^{\eta}\min\\{1,(\beta\log
X)^{-1}\\}(1+\beta\log X)^{1-A}d\beta.$

Splitting the $\beta$-integral at $1/\log X$ and evaluating, we obtain

$\displaystyle T_{2}$
$\displaystyle\ll_{A,B,C}(\log(ew))\mathcal{P}_{f}(X)\left(\frac{1}{\log
X}+\frac{1_{A=1}\log(\eta\log X)+1}{(\log X)^{A}}\left((\log
X)^{A-1}+\eta^{1-A}\right)\right)$
$\displaystyle\ll_{B}\mathcal{P}_{f}(X)\left(\frac{(\log(ew))(1+1_{A=1}\log(\log
X/\log(ew)))}{\log X}+\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{A}\right)$
$\displaystyle\ll\mathcal{P}_{f}(X)\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\min\\{1,A\\}}\left(1+1_{A=1}\log\left(\frac{\log
X}{\log(ew)}\right)\right).$

We now turn to $T_{1}$. By evaluating the $\alpha$ integral, we have

$T_{1}\ll\frac{1}{\log X}\int_{0}^{\eta}\min\\{\log X,1/\beta\\}\max_{|t|\leq
T}\frac{|\mathcal{F}(c_{0}+\beta+it)||1-w^{-\beta-
it}|}{|c_{0}+\beta+it|}d\beta.$

Put $T^{\prime}:=\frac{1}{2}(\log X)^{B}$. If the maximum occurs at
$|t|>T^{\prime}$ then using the second estimate in Lemma 3.7 we obtain

$T_{1}\ll_{B,C}\frac{1}{\log X}\int_{0}^{\eta}\min\\{\log
X,1/\beta\\}\cdot\frac{(\log X)^{B}}{T^{\prime}}d\beta\ll\frac{1}{\log
X}\left((\log X)\cdot\frac{1}{\log X}+\log\left(\eta\log
X\right)\right)\ll\frac{\log\log X}{\log X}.$

Thus, suppose the maximum occurs with $|t|\leq T^{\prime}$. Applying [7, Lem.
3.1], we get

$\displaystyle\max_{|t|\leq
T^{\prime}}|\mathcal{F}(c_{0}+\beta+it)(1-w^{-\beta-it})|$
$\displaystyle\leq\max_{|t|\leq(\log
X)^{B}}|\mathcal{F}(c_{0}+it)(1-w^{-it})|+O\left(\frac{\beta}{(\log
X)^{B}}\sum_{n\leq X}\frac{|f(n)|}{n^{1+1/\log X}}\right)$
$\displaystyle=\max_{|t|\leq(\log
X)^{B}}|\mathcal{F}(c_{0}+it)(1-w^{-it})|+O_{B,C}\left(\frac{\beta}{(\log
X)^{B-1}}\mathcal{P}_{f}(X)\right).$

Inserting this into the $\beta$ integral yields, in this case,

$\displaystyle T_{1}$ $\displaystyle\ll_{B}\max_{|t|\leq(\log
X)^{B}}|\mathcal{F}(c_{0}+it)(1-w^{-it})|\cdot\frac{1}{\log
X}\int_{0}^{\eta}\min\\{\log
X,1/\beta\\}d\beta+\frac{\mathcal{P}_{f}(X)}{(\log
X)^{B}}\int_{0}^{\eta}\beta\min\\{\log X,1/\beta\\}d\beta$
$\displaystyle\ll_{B}\max_{|t|\leq(\log X)^{B}}|\mathcal{F}(1+1/\log
X+it)(1-w^{-it})|\cdot\frac{\log(\log X/\log(ew))}{\log
X}+\frac{\mathcal{P}_{f}(X)}{(\log X)^{B}}.$

Finally, we focus on the maximum here. Note that
$|1-w^{-it}|\ll\min\\{1,|t|\log(ew)\\}$, so combining Lemma 3.7 with our
hypothesis (8), we obtain

$\displaystyle\max_{|t|\leq(\log X)^{B}}|\mathcal{F}(1+1/\log
X+it)(1-w^{-it})|$ $\displaystyle\ll_{A,B,C}(\log
X)\mathcal{P}_{f}(X)\cdot\max_{|t|\leq(\log
X)^{B}}\min\\{1,|t|\log(ew)\\}e^{-\rho(f,n^{i(t_{0}+t)};X)^{2}}$
$\displaystyle\ll_{A,B}(\log X)\mathcal{P}_{f}(X)\max_{|t|\leq(\log
X)^{B}}\min\\{1,|t|\log(ew)\\}\left(\frac{1}{(\log
X)^{\sigma}}+\frac{1}{(1+|t|\log X)^{\sigma}}\right)$ $\displaystyle\ll(\log
X)\mathcal{P}_{f}(X)\cdot\left(\frac{\log(ew)}{\log
X}\right)^{\min\\{1,\sigma\\}}.$

Hence, as $\hat{\sigma}=\min\\{1,\sigma\\}\leq A\leq B$, and $(\log
X)^{-1}\ll_{A}(\log X)^{-A}\mathcal{P}_{f}(X)$ by (7) we get

$\displaystyle T_{1}$ $\displaystyle\ll_{A,B,C}\log\left(\frac{\log
X}{\log(ew)}\right)\left(\frac{\log(ew)}{\log
X}\right)^{\hat{\sigma}}\mathcal{P}_{f}(X)+\frac{\mathcal{P}_{f}(X)}{(\log
X)^{B}}+\frac{\log\log X}{\log X}$ $\displaystyle\ll\log\left(\frac{\log
X}{\log(ew)}\right)\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\hat{\sigma}}\mathcal{P}_{f}(X).$

Combining all of these bounds and inserting them into (17), we thus find that

$\displaystyle\left|\frac{1}{X}\sum_{n\leq
X}f(n)n^{-it_{0}}-\frac{w}{X}\sum_{n\leq X/w}f(n)n^{-it_{0}}\right|$
$\displaystyle\ll_{A,B,C}\mathcal{P}_{f}(X)\log\left(\frac{\log
X}{\log(ew)}\right)\left(\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\min\\{1,A\\}}+\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\hat{\sigma}}\right)$
$\displaystyle\ll\mathcal{P}_{f}(X)\log\left(\frac{\log
X}{\log(ew)}\right)\left(\frac{\log(ew)+\log\log X}{\log
X}\right)^{\hat{\sigma}}.$

This proves the first claim.
The second claim can be deduced similarly, since (using the same notation as
above) in the first step we have (after shifting $s\mapsto s-it_{0}$)

$\displaystyle\frac{1}{X^{1+it_{0}}}\sum_{n\leq
X}f(n)-\left(\frac{w}{X}\right)^{1+it_{0}}\sum_{n\leq X/w}f(n)$
$\displaystyle=\int_{0}^{\eta}\int_{0}^{\eta}\frac{1}{\pi
i}\int_{c_{0}-i\infty}^{c_{0}+i\infty}\mathcal{S}(s-\alpha-\beta)\mathcal{L}(s+\beta)\frac{\mathcal{L}^{\prime}}{\mathcal{L}}(s+\alpha)\frac{\mathcal{L}^{\prime}}{\mathcal{L}}(s+\alpha+2\beta)\frac{X^{s-1}(1-w^{1-s})}{s+it_{0}}dsd\beta
d\alpha$ $\displaystyle+O_{A,B,C}\left(\left(\frac{\log y}{\log
X}\right)^{A}\mathcal{P}_{f}(X)\right),$

which simply localizes the argument above to the range $|t+t_{0}|\leq T$
instead. ∎

Theorem 3.6 may be directly applied to obtain the first estimate in Theorem
2.2. To obtain the second, we will use the following corollary of Theorem 3.6
that allows us to pass from $n^{-it_{0}}$-twisted sums to untwisted sums of
$f(n)$ on long intervals.

######  Corollary 3.9.

Let $t_{0}=t_{0}(f,X)$ be as above. Then for any $x\in(X/2,X]$,

$\frac{1}{x}\sum_{n\leq
x}f(n)n^{-it_{0}}=\frac{1+it_{0}}{x^{1+it_{0}}}\sum_{n\leq
x}f(n)+O_{A,B,C}\left(|t_{0}|\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right).$

###### Proof.

By partial summation, we have

(18) $\frac{1}{x}\sum_{n\leq
x}f(n)n^{-it_{0}}=\frac{1}{x}\int_{1}^{x}u^{-it_{0}}d\\{\sum_{n\leq
x}f(n)\\}=\frac{1}{x^{1+it_{0}}}\sum_{n\leq
x}f(n)+\frac{it_{0}}{x}\int_{1}^{x}\frac{1}{u^{1+it_{0}}}\sum_{n\leq
u}f(n)du.$

We split the integral over $u$ at $x/(\log X)$. In the first range we use the
trivial bound together with Lemma 3.2, obtaining

$\leq\frac{|t_{0}|}{x}\int_{1}^{x/(\log X)^{2}}\left(\frac{1}{u}\sum_{n\leq
u}|f(n)|\right)du\ll_{B,C}\frac{|t_{0}|}{(\log X)^{2}}\prod_{p\leq
X}\left(1+\frac{|f(p)|}{p}\right)\ll\frac{|t_{0}|}{\log X}\mathcal{P}_{f}(X).$

In the remaining range $x/(\log X)^{2}<u\leq x$ we apply the second claim in
Theorem 3.6 (with $1\leq w\leq(\log X)^{2}$), which gives

$\displaystyle\frac{it_{0}}{x}\int_{x/(\log
X)^{2}}^{X}\left(\frac{1}{x^{1+it_{0}}}\sum_{n\leq
x}f(n)+O_{A,B,C}\left(\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right)\right)du$
$\displaystyle=\left(\frac{1}{x^{1+it_{0}}}\sum_{n\leq
x}f(n)\right)\cdot\frac{it_{0}}{x}\int_{x/(\log
X)^{2}}^{x}du+O_{A,B,C}\left(|t_{0}|\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right)$
$\displaystyle=\frac{it_{0}}{x^{1+it_{0}}}\sum_{n\leq
x}f(n)+O_{A,B,C}\left(|t_{0}|\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right).$

Combining this with the estimate from $1\leq u\leq x/(\log X)^{2}$, then
inserting this into (18), we prove the claim. ∎

###### Proof of Theorem 2.2.

We begin by proving the first estimate in the statement of the theorem. Let
$X/(\log X)^{\hat{\sigma}/2}\leq h\leq X/10$. Note that by writing $x-h=x/w$,
where $w:=(1-h/x)^{-1}\in[1,2]$, the LHS is

(19) $\frac{1}{h}\left(\sum_{n\leq x}f(n)n^{-it_{0}}-\sum_{n\leq
x/w}f(n)n^{-it_{0}}\right).$

and similarly the main term in the RHS is

(20) $\frac{2}{X}\left(\sum_{n\leq X}f(n)n^{-it_{0}}-\sum_{n\leq
X/2}f(n)n^{-it_{0}}\right).$

By Theorem 3.6, (19) becomes

$\displaystyle\frac{1}{h}\left(\left(1-\frac{1}{w}\right)\sum_{n\leq
x}f(n)n^{-it_{0}}\right)+O_{A,B,C}\left(\frac{X}{h}\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\mathcal{P}_{f}(X)\right)$
$\displaystyle=\frac{1}{x}\sum_{n\leq
x}f(n)n^{-it_{0}}+O_{A,B,C}\left(\frac{(\log\log X)^{\hat{\sigma}+1}}{(\log
X)^{\hat{\sigma}/2}}\mathcal{P}_{f}(X)\right).$

Similarly, applying Theorem 3.6 twice to (20), we also find that

$\displaystyle\frac{2}{X}\sum_{X/2<n\leq X}f(n)n^{-it_{0}}$
$\displaystyle=\frac{2}{X}\left(1-\frac{1}{2}\right)\sum_{n\leq
X}f(n)n^{-it_{0}}+O_{A,B,C}\left(\frac{(\log\log X)^{\hat{\sigma}+1}}{(\log
X)^{\hat{\sigma}}}\mathcal{P}_{f}(X)\right)$ (21)
$\displaystyle=\frac{1}{x}\sum_{n\leq
x}f(n)n^{-it_{0}}+O_{A,B,C}\left(\frac{(\log\log X)^{\hat{\sigma}+1}}{(\log
X)^{\hat{\sigma}}}\mathcal{P}_{f}(X)\right),$

viewing $x=X/u$ for some $u\in[1,2]$ in the last step. Combined with the
previous estimate, we deduce the first claimed estimate of the theorem.

To prove the second claimed estimate we apply Corollary 3.9 to obtain

$\displaystyle\sum_{x-h<n\leq x}f(n)=\frac{x^{it_{0}}}{1+it_{0}}\sum_{n\leq
x}f(n)n^{-it_{0}}-\frac{(x-h)^{it_{0}}}{1+it_{0}}\sum_{n\leq
x-h}f(n)n^{-it_{0}}+O\left(X\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right).$

Setting $w:=(1-h/x)^{-1}$ as in (19), we have

$\displaystyle\frac{x^{it_{0}}}{1+it_{0}}\sum_{n\leq
x}f(n)n^{-it_{0}}-\frac{(x-h)^{it_{0}}}{1+it_{0}}\sum_{n\leq
x-h}f(n)n^{-it_{0}}$
$\displaystyle=\left(\frac{x^{1+it_{0}}}{1+it_{0}}-\frac{1}{w}\frac{(x-h)^{it_{0}}}{1+it_{0}}\right)\sum_{n\leq
x}f(n)n^{-it_{0}}+O\left(X\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right)$
$\displaystyle=\frac{x^{it_{0}}-(x-h)^{1+it_{0}}}{1+it_{0}}\cdot\frac{1}{x}\sum_{n\leq
x}f(n)n^{-it_{0}}+O\left(X\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right).$

By (21), the main term here is

$\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
X}f(n)n^{-it_{0}}+O\left(h\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right).$

Combining these estimates, we thus obtain

$\frac{1}{h}\sum_{x-h<n\leq
x}f(n)=\frac{1}{h}\int_{x-h}^{x}u^{it_{0}}du\cdot\frac{2}{X}\sum_{X/2<n\leq
x}f(n)n^{-it_{0}}+O\left(\frac{X}{h}\mathcal{P}_{f}(X)\frac{(\log\log
X)^{\hat{\sigma}+1}}{(\log X)^{\hat{\sigma}}}\right),$

and so the second claimed estimate of the theorem follows from $h\geq X/(\log
X)^{\hat{\sigma}/2}$. ∎

##  4\. Applying the Matomäki-Radziwiłł Method

In this section, which broadly follows the lines of the proof of [26, Thm. 3],
we set out the key elements of the proof of Theorem 2.1.

###  4.1. Large sieve estimates

The content of this subsection can essentially all be found in [26, Sec. 6 and
7] and in [24, Sec. 3]. In what follows, a set $\mathcal{T}\subset\mathbb{R}$
is said to be _well-spaced_ if $|t_{1}-t_{2}|\geq 1$ for any distinct
$t_{1},t_{2}\in\mathcal{T}$.

######  Lemma 4.1 (Sparse large sieve for multiplicative sequences).

Let $T\geq 1$ and $2\leq N\leq X$. Let $\\{a_{n}\\}_{n\leq N}$ be a sequence
of complex numbers. Let $\mathcal{T}\subset[-T,T]$ be well-spaced. The
following bounds hold:

  1. (a)

($L^{2}$ mean value theorem, sparse version)

$\displaystyle\int_{-T}^{T}|\sum_{n\leq N}a_{n}n^{-it}|^{2}dt$
$\displaystyle\ll T\sum_{n\leq N}|a_{n}|^{2}+T\sum_{n\leq N}\sum_{1\leq|m|\leq
n/T}|a_{n}a_{m+n}|.$

  2. (b)

($L^{2}$ mean value theorem with multiplicative majorant) Let $1\leq M\leq N$,
and let $c>0$. Assume there is a multiplicative function
$f:\mathbb{N}\rightarrow\mathbb{C}$ satisfying $|f(n)|\leq d_{B}(n)^{C}$ such
that $|a_{n}|\leq c|f(n)|$ for all $n\leq N$. Then

$\displaystyle\int_{-T}^{T}|\sum_{N-M<n\leq N}\frac{a_{n}n^{-it}}{n}|^{2}$
$\displaystyle\ll_{B,C}c^{2}\left(\frac{TM}{N^{2}}\mathcal{P}_{f^{2}}(N)+\frac{M}{N}\mathcal{P}_{f}(N)^{2}\right).$

  3. (c)

(Discrete mean value theorem)

$\sum_{t\in\mathcal{T}}|\sum_{N/3<n\leq
N}\frac{a_{n}n^{-it}}{n}|^{2}\ll\min\left\\{\left(1+\frac{T}{N}\right)\log(2N),\left(1+|\mathcal{T}|\frac{T^{1/2}}{N}\right)\log(2T)\right\\}\frac{1}{N}\sum_{N/3<n\leq
N}|a_{n}|^{2}.$

###### Proof.

Part (a) is [24, Lem. 3.2], part (b) is proven in the same way as777The
technique used there relies on the main result of [14], which is valid
generally for multiplicative functions that are bounded by a power of the
divisor function. [24, Lem. 3.4] and part (c) is a combination of [26, Lem. 7
and 9]. ∎

######  Lemma 4.2 (Large Sieve with Prime Support).

Let $B,T\geq 1$, $P\geq 10$. Let $\\{a_{p}\\}_{P<p\leq 2P}$ be a sequence with
$\max_{P<p\leq 2P}|a_{p}|\leq B$. Let $P(s):=\sum_{P<p\leq 2P}a_{p}p^{-s}$,
for $s\in\mathbb{C}$ and let $\mathcal{T}\subset[-T,T]$ be a well-spaced set.

  1. (a)

(Halász-Montgomery estimate for primes)

$\sum_{t\in\mathcal{T}}|P(1+it)|^{2}\ll_{B}\frac{1}{(\log
P)^{2}}\left(1+|\mathcal{T}|(\log T)^{2}\exp\left(-\frac{\log P}{(\log
T)^{2/3+\varepsilon}}\right)\right).$

  2. (b)

(Large values estimate) If $\mathcal{T}$ consists only of $t\in[-T,T]$ with
$|P(1+it)|\geq V^{-1}$ then

$|\mathcal{T}|\ll_{B}T^{2\frac{\log V}{\log P}}V^{2}\exp\left(2B\frac{\log
T}{\log P}\log\log T\right).$

###### Proof.

Part (a) is [26, Lem. 11], while part (b) is proven precisely as in [26, Lem.
8], keeping track of the upper bound condition $|a_{p}|\leq B$. ∎

###  4.2. Dirichlet Polynomial Decomposition

The following is a variant of [26, Lem. 12] tailored to elements of
$\mathcal{M}(X;A,B,C;\gamma,\sigma)$.

######  Lemma 4.3.

Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Let $T\geq 1$, $1\leq H\leq
X^{1/2}$ and $1\leq P\leq Q\leq X^{1/2}$. Set $\mathcal{I}$ to be the interval
of integers $\left\lfloor H\log P\right\rfloor\leq v\leq H\log Q$, and let
$\mathcal{T}\subset[-T,T]$. Then

$\displaystyle\int_{\mathcal{T}}$
$\displaystyle\left|\sum_{\begin{subarray}{c}X/3<n\leq X\\\
\omega_{[P,Q]}(n)\geq
1\end{subarray}}\frac{f(n)}{n^{1+it}}\right|^{2}dt\ll_{B,C}H\log(Q/P)\sum_{v\in\mathcal{I}}\int_{\mathcal{T}}|Q_{v,H}(1+it)R_{v,H}(1+it)|^{2}dt$
$\displaystyle+\left(\frac{1}{H}+\frac{1}{P}\right)\left(\frac{T}{X}\mathcal{P}_{f^{2}}(X)+\mathcal{P}_{f}(X)^{2}\right),$

where for $v\in\mathcal{I}$ and $s\in\mathbb{C}$ we have set

$\displaystyle Q_{v,H}(s)$ $\displaystyle:=\sum_{\begin{subarray}{c}P\leq
p\leq Q\\\ v/H\leq\log p\leq(v+1)/H\end{subarray}}f(p)p^{-s}$ $\displaystyle
R_{v,H}(s)$ $\displaystyle:=\sum_{Xe^{-v/H}/3\leq m\leq
Xe^{-v/H}}\frac{f(m)}{m^{s}(\omega_{[P,Q]}(m)+1)}.$

###### Proof.

The proof is the same as that of [26, Lem. 12] (with
$a_{n}=f(n)1_{\omega_{[P,Q]}\geq 1}(n)$, $b_{m}=f(m)$ and $c_{p}=f(p)$ for
$P\leq p\leq Q$), with appropriate appeal to Lemma 4.1 in place of the usual
mean value theorem. For example (as on [26, top of p.20]), for
$Y\in\\{X/(3Q),X/P\\}$ we have

$\displaystyle\int_{\mathcal{T}}\left|\sum_{m\in[Ye^{-1/H},Ye^{1/H}]}f(m)m^{-1-it}\right|^{2}dt$
$\displaystyle\ll_{B,C}\frac{T}{XH}\mathcal{P}_{f^{2}}(X)+\frac{1}{H}\mathcal{P}_{f}(X)^{2}.$

∎

######  Lemma 4.4.

Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Let $T,Y_{1},Y_{2}\geq 1$,
$1\leq X^{\prime}\leq X/Y_{1}$ and let $\ell:=\lceil\frac{\log Y_{2}}{\log
Y_{1}}\rceil$. Define

$Q(s):=\sum_{Y_{1}/2<p\leq Y_{1}}c_{p}p^{-s},\quad\quad
A(s):=\sum_{X^{\prime}/(2Y_{2})<m\leq X^{\prime}/Y_{2}}f(m)m^{-s},$

where $|c_{p}|\leq B$ for all $Y_{1}/2<p\leq Y_{1}$. Finally, let
$\mathcal{T}\subseteq[-T,T]$. Then

$\int_{\mathcal{T}}|Q(1+it)^{\ell}A(1+it)|^{2}dt\ll_{B,C}B^{2\ell}(\ell!)^{2}\left(\frac{T}{X}\mathcal{P}_{f^{2}}(X)+\mathcal{P}_{f}(X)^{2}\right).$

###### Proof.

Writing $c_{p}:=Bc_{p}^{\prime}$, where now $|c_{p}^{\prime}|\leq 1$ and
letting $\tilde{Q}(s)$ denote the Dirichlet polynomial with $c_{p}$ replaced
by $c_{p}^{\prime}$ for all $p\in(Y_{1}/2,Y_{1}]$, the LHS in the statement is

$\ll B^{2\ell}\int_{\mathcal{T}}|\tilde{Q}(1+it)A(1+it)|^{2}dt.$

The rest of the proof is essentially the same as that of [24, Lem. 7.1], save
that the function $g^{\ast}$ is replaced by $|f|$ (this does not affect the
proof, which depends on our Lemma 3.2 and [14, Thm. 3], both of which also
apply to $|f|$.) ∎

###  4.3. Integral Averages of Dirichlet Polynomials

As above, we write $t_{0}=t_{0}(f,X)$ to denote an element of $[-X,X]$ that
minimizes the map

$t\mapsto\sum_{p\leq
X}\frac{|f(p)|-\text{Re}(f(p)p^{-it})}{p}=\rho(f,n^{it};X)^{2}.$

######  Proposition 4.5.

Let $1/\log X\leq\delta\leq 1$, and put
$I_{\delta}:=[t_{0}-\delta^{-1},t_{0}+\delta^{-1}]$. Let $\\{a_{n}\\}_{n\leq
X}$ be a sequence of complex numbers. Then

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{\begin{subarray}{c}x-h<m\leq
x\end{subarray}}a_{m}-\frac{1}{2\pi}\int_{I_{\delta}}A(1+it)\frac{x^{1+it}-(x-h)^{1+it}}{h(1+it)}dt\right|^{2}dx$
$\displaystyle\ll\int_{[-X/h,X/h]\backslash
I_{\delta}}|A(1+it)|^{2}dt+\max_{T\geq
X/h}\frac{X}{hT}\int_{T}^{2T}|A(1+it)|^{2}dt,$

where we have set

$A(s):=\sum_{\begin{subarray}{c}X/3<n\leq
X\end{subarray}}\frac{a(n)}{n^{s}},\quad\quad s\in\mathbb{C}.$

###### Proof.

For each $x\in[X/2,X]$, Perron’s formula gives

$\frac{1}{h}\sum_{x-h<m\leq
x}a_{m}=\frac{1}{2\pi}\int_{\mathbb{R}}A(1+it)\frac{x^{1+it}-(x-h)^{1+it}}{h(1+it)}dt.$

Subtracting the contribution from $|t-t_{0}|\leq\delta^{-1}$, the LHS in the
statement becomes

$\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{2\pi}\int_{\mathbb{R}\backslash
I_{\delta}}A(1+it)\frac{x^{1+it}-(x-h)^{1+it}}{h(1+it)}dt\right|^{2}dx.$

The remainder of the proof is identical to that of [26, Lemma 14] (which only
uses the boundedness of the coefficients $\\{a_{n}\\}_{n}$ there for the
corresponding contribution of $I_{\delta}$). ∎

###  4.4. Restricting to a “nicely factored” set

We fix parameters $\eta\in(0,1/12)$, $Q_{1}:=h_{0}$, $P_{1}:=(\log
h_{0})^{40B/\eta}$ and $P_{j}:=\exp\left(j^{4j-2}(\log Q_{1})^{j-1}(\log
P_{1})\right)$, $Q_{j}:=\exp\left(j^{4j}(\log Q_{1})^{j}\right)$, for $1\leq
j\leq J$, where $J$ is maximal with $Q_{J}\leq\exp\left(\sqrt{\log X}\right)$.
We highlight the different choice of $P_{1}$, with all other choices being the
same as in [26, Sec. 2 and 8]. We also let

$\mathcal{S}=\mathcal{S}_{X,P_{1},Q_{1}}:=\\{n\leq
X:\omega_{[P_{j},Q_{j}]}(n)\geq 1\text{ for all }1\leq j\leq J\\}.$

######  Remark 4.6.

The following properties may be verified directly, as long as $h_{0}$ is
sufficiently large (in terms of $B$):

  1. (1)

$\log P_{j}\geq\frac{8Bj^{2}}{\eta}\log\log(2BQ_{j+1})\text{ for all $1\leq
j\leq J$}$

  2. (2)

$\log Q_{j}\leq 2^{4j}(\log Q_{j-1})(\log Q_{1})\leq(\log Q_{j-1})^{3}\leq
Q_{j-1}^{1/24}$

  3. (3)

$\frac{\log P_{j}}{\log Q_{j}}=\frac{\log P_{1}}{j^{2}\log Q_{1}}$ for $2\leq
j\leq J$, so the terms $\\{\log P_{j}/\log Q_{j}\\}_{j\geq 1}$ are summable.

We will use these in due course.

We wish to reduce our work to handling short and long averages with $n$
restricted to the set $\mathcal{S}$. To handle averages of $f(n)$ for
$n\notin\mathcal{S}$ we use the following result. For a set of integers
$\mathcal{A}$, we write $(n,\mathcal{A})=1$ to mean that $(n,a)=1$ for all
$a\in\mathcal{A}$.

######  Lemma 4.7.

Let $1<P\leq Q\leq X$, and let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Then
for any $10\leq h_{0}\leq X/10H(f;X)$ and $h:=h_{0}H(f;X)$,

$\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{\begin{subarray}{c}x-h<m\leq
x\\\
(m,[P,Q])=1\end{subarray}}f(m)\right|^{2}dx\ll_{A,B,C}\left(\left(\frac{\log
P}{\log Q}\right)^{A}+\frac{1}{h_{0}}\right)\mathcal{P}_{f}(X)^{2}.$

In particular,

$\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{\begin{subarray}{c}x-h<m\leq
x\\\
m\notin\mathcal{S}\end{subarray}}f(m)\right|^{2}dx\ll_{A,B,C}\left(\frac{\log\log
h_{0}}{\log h_{0}}\right)^{A}\mathcal{P}_{f}(X)^{2}.$

###### Proof.

Expanding the square and applying the triangle inequality, the LHS is

$\displaystyle\leq\frac{2}{h^{2}X}\sum_{\begin{subarray}{c}X/2-h<m_{1},m_{2}\leq
X\\\ |m_{1}-m_{2}|\leq h\\\
(m_{1}m_{2},[P,Q])=1\end{subarray}}|f(m_{1})\overline{f(m_{2})}|\int_{X/2}^{X}1_{[m_{1},m_{1}+h)}(x)1_{[m_{2},m_{2}+h)}(x)dx$
$\displaystyle\ll\frac{1}{hX}\sum_{\begin{subarray}{c}X/3<m\leq
X\end{subarray}}|f(m)|^{2}+\frac{1}{hX}\sum_{1\leq|l|\leq
h}\sum_{\begin{subarray}{c}X/3<m\leq X\\\
(m,[P,Q])=1\end{subarray}}|f(m)||f(m+l)|.$

By Lemma 3.2 and (11), the first term on the RHS is bounded as

(22)
$\ll_{B,C}\frac{1}{h_{0}H(f;X)}\mathcal{P}_{f^{2}}(X)\ll_{B}\frac{1}{h_{0}}\mathcal{P}_{f}(X)^{2}.$

Next, to bound the correlation sums we apply888This result is stated in [24]
for bounded multiplicative functions, but the proof there works identically
for divisor-bounded functions as well since it relies principally on the
general setup of [14, Thm. 3]. [24, Lem. 3.3] (with $r_{1}=r_{2}=1$) to the
pair of multiplicative functions $f1_{\mathcal{S}^{c}}$ and $f$, which gives

$\sum_{1\leq|l|\leq h}\sum_{\begin{subarray}{c}X/3<n\leq X\\\
(n,[P,Q])=1\end{subarray}}|f(n)f(n+l)|\ll_{B,C}hX\mathcal{P}_{f1_{(m,[P,Q])=1}}(X)\mathcal{P}_{f}(X).$

Since $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$, by (7) we get

$\displaystyle\mathcal{P}_{f1_{(m,[P,Q])=1}}(X)\asymp_{B}\mathcal{P}_{f}(X)\prod_{P\leq
p\leq Q}\left(1+\frac{1-|f(p)|}{p}\right)\left(1-\frac{1}{p}\right)$
$\displaystyle\ll_{A,B}\mathcal{P}_{f}(X)\exp\left(A\frac{\log P}{\log
Q}\right)$ $\displaystyle=\left(\frac{\log P}{\log
Q}\right)^{A}\mathcal{P}_{f}(X).$

It follows that

$\frac{1}{hX}\sum_{1\leq|l|\leq h}\sum_{\begin{subarray}{c}X/3<n\leq X\\\
n\notin\mathcal{S}\end{subarray}}|f(n)||f(n+l)|\ll_{A,B}\left(\frac{\log
P}{\log Q}\right)^{A}\mathcal{P}_{f}(X)^{2}.$

The first claim now follows upon combining this with (22). The second claim
follows similarly, save that in the argument above the term
$\mathcal{P}_{f1_{(m,[P,Q])=1}}(X)$ is replaced by

$\displaystyle\mathcal{P}_{f1_{\mathcal{S}^{c}}}(X)$
$\displaystyle\ll_{B}\mathcal{P}_{f}(X)\prod_{1\leq j\leq J}\prod_{P_{j}\leq
p\leq
Q_{j}}\left(1+\frac{1-|f(p)|}{p}\right)\left(1-\frac{1}{p}\right)\ll_{A,B}\mathcal{P}_{f}(X)\exp\left(A\sum_{j\geq
1}\frac{\log P_{j}}{\log Q_{j}}\right)$ $\displaystyle\ll\left(\frac{\log
P_{1}}{\log Q_{1}}\right)^{A}\mathcal{P}_{f}(X)\ll_{A,B}\left(\frac{\log\log
h_{0}}{\log h_{0}}\right)^{A}\mathcal{P}_{f}(X).$

∎

Having disposed of $n\notin\mathcal{S}$, we now concerntrate on
$n\in\mathcal{S}$. To prove Theorem 2.1 we will apply Proposition 4.5 to the
sequence $a_{m}=f(m)1_{m\in\mathcal{S}}$, in combination with the following
key proposition.
Recall that $\kappa:=\frac{\min\\{1,\sigma\\}}{8B+21}$. We also define
$\Delta:=(2B+5)\kappa$, and

$F(s):=\sum_{\begin{subarray}{c}X/3<n\leq X\\\
n\in\mathcal{S}\end{subarray}}\frac{f(n)}{n^{s}},s\in\mathbb{C}.$

######  Proposition 4.8.

Set $\delta=(\log X)^{-\Delta}$. Then

$\int_{[-X/h,X/h]\backslash
I_{\delta}}|F(1+it)|^{2}dt\ll_{A,B,C}\left(\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-2\eta}}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\mathcal{P}_{f}(X)^{2}.$

######  Remark 4.9.

We remark that both terms in Proposition 4.5 can be treated using Proposition
4.8. Indeed, by Lemma 4.1(b),

$\frac{1}{T}\int_{T}^{2T}|F(1+it)|^{2}dt\ll_{B,C}\frac{1}{X}\mathcal{P}_{f^{2}}(X)+\frac{1}{T}\mathcal{P}_{f}(X)^{2}.$

and therefore

$\max_{T\geq X(\log
h_{0})^{A}/h}\frac{X/h}{T}\int_{T}^{2T}|F(1+it)|^{2}dt\ll_{B,C}\frac{1}{h}\mathcal{P}_{f^{2}}(X)+\frac{1}{(\log
h_{0})^{A}}\mathcal{P}_{f}(X)^{2}\ll\frac{1}{(\log
h_{0})^{A}}\mathcal{P}_{f}(X)^{2},$

which is obviously sufficient in Theorem 2.1. We clearly also have

$\max_{X/h<T\leq X(\log
h_{0})^{A}/h}\frac{X/h}{T}\int_{T}^{2T}|F(1+it)|^{2}dt\leq\int_{X/h}^{X(\log
h_{0})^{A}/h}|F(1+it)|^{2}dt.$

This expression will also be bounded using Proposition 4.8 with $h$ replaced
by $h/(\log h_{0})^{A}$, which does not change the form of the final
estimates.

###  4.5. Proof of Proposition 4.8

The proof follows the same lines as those in [26, Sec. 8], save that we apply
our versions of the corresponding lemmas that address the growth of
$f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. We sketch the details here,
emphasizing the main differences.
We select parameters $\alpha_{1},\ldots,\alpha_{J}$ such that
$\alpha_{j}:=\frac{1}{4}-\eta\left(1+\frac{1}{2j}\right)$. For each $1\leq
j\leq J$ let

$\mathcal{I}_{j}:=[\left\lfloor H_{j}\log P_{j}\right\rfloor,H_{j}\log
Q_{j}]\cap\mathbb{Z},\quad\quad H_{j}:=j^{2}\frac{P_{1}^{1/6-\eta}}{(\log
Q_{1})^{1/3}}.$

Also, with $s\in\mathbb{C}$ we let

$Q_{v,H_{j}}(s):=\sum_{\begin{subarray}{c}P_{j}\leq p\leq Q_{j}\\\
e^{v/H_{j}}\leq p\leq e^{(v+1)/H_{j}}\end{subarray}}f(p)p^{-s},\quad\quad
R_{v,H_{j}}(s):=\sum_{\frac{1}{3}Xe^{-v/H_{j}}<Xe^{-v/H_{j}}}\frac{f(m)}{m^{s}(1+\omega_{[P_{j},Q_{j}]}(m))}.$

We split the set of $t\in\mathcal{X}:=[-X/h,X/h]\backslash I_{\delta}$ into
sets

$\displaystyle\mathcal{T}_{1}$
$\displaystyle:=\\{t\in\mathcal{X}:|Q_{v,H_{1}}(1+it)|\leq
e^{-\alpha_{1}v/H_{1}}\text{ for all }v_{1}\in\mathcal{I}_{1}\\}$
$\displaystyle\mathcal{T}_{j}$
$\displaystyle:=\\{t\in\mathcal{X}:|Q_{v,H_{j}}(1+it)|\leq
e^{-\alpha_{j}v/H_{j}}\text{ for all
}v_{j}\in\mathcal{I}_{j}\\}\backslash\bigcup_{1\leq i\leq
j-1}\mathcal{T}_{i},\quad 2\leq j\leq J,\text{ if }J\geq 2$
$\displaystyle\mathcal{U}$ $\displaystyle:=\mathcal{X}\backslash\bigcup_{1\leq
j\leq J}\mathcal{T}_{j}.$

We thus have

$\int_{[-X/h,X/h]\backslash I_{\delta}}|F(1+it)|^{2}dt=\sum_{1\leq j\leq
J}\int_{\mathcal{T}_{j}}|F(1+it)|^{2}dt+\int_{\mathcal{U}}|F(1+it)|^{2}dt.$

We estimate the contributions from $\mathcal{T}_{j}$ as in [26], save that in
the applications of the large sieve inequalities we use Lemma 4.1(b); as an
example we will give full details for $j=1$ and highlight the main changes for
the corresponding bounds for $2\leq j\leq J$.
In each case we apply Lemma 4.3 to obtain

$\int_{\mathcal{T}_{j}}|F(1+it)|^{2}dt\ll_{B,C}\mathcal{M}_{j}+\mathcal{E}_{j},$

where we set

$\displaystyle\mathcal{M}_{j}$
$\displaystyle:=H_{j}\log(Q_{j}/P_{j})\sum_{v\in\mathcal{I}_{j}}\int_{\mathcal{T}_{j}}|Q_{v,H_{j}}(1+it)R_{v,H_{j}}(1+it)|^{2}dt$
$\displaystyle\mathcal{E}_{j}$
$\displaystyle:=\left(\frac{1}{H_{j}}+\frac{1}{P_{j}}\right)\left(\frac{T}{X}\mathcal{P}_{f^{2}}(X)+\mathcal{P}_{f}(X)^{2}\right).$

The choice of parameters gives, by (22),

(23) $\sum_{1\leq j\leq J}\mathcal{E}_{j}\ll\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}\left(\frac{1}{h_{0}H(f;X)}\mathcal{P}_{f^{2}}(X)+\mathcal{P}_{f}(X)^{2}\right)\ll_{B,C}\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}\mathcal{P}_{f}(X)^{2},$

since $\sum_{j}P_{j}^{-1}\ll P_{1}^{-1}$ and
$H(f;X)^{-1}\mathcal{P}_{f^{2}}(X)\ll_{B,C}\mathcal{P}_{f}(X)^{2}$.
We next consider the contribution from the main terms. When $j=1$, since
$v/H_{1}\leq\log Q_{1}=\log h_{0}$, we have

$\displaystyle\mathcal{M}_{1}$ $\displaystyle\leq H_{1}\log
Q_{1}\sum_{v\in\mathcal{I}_{1}}e^{-2\alpha_{1}v/H_{1}}\int_{-X/h}^{X/h}|R_{v,H_{j}}(1+it)|^{2}dt$
$\displaystyle\ll_{B,C}H_{1}\log
Q_{1}\sum_{v\in\mathcal{I}_{1}}e^{-2\alpha_{1}v/H_{1}}\left(\mathcal{P}_{f}(X)^{2}+\frac{e^{v/H_{1}}}{h_{0}H(f;X)}\mathcal{P}_{f^{2}}(X)\right)$
$\displaystyle\ll_{B,C}\frac{H_{1}^{2}\log
Q_{1}}{P_{1}^{2\alpha_{1}}}\mathcal{P}_{f}(X)^{2},$

treating $\mathcal{P}_{f^{2}}(X)$ using (22). Since $H_{1}^{2}\log
Q_{1}=P_{1}^{1/3-2\eta}(\log Q_{1})^{1/3}\leq P_{1}^{1/3-\eta}$,
$P_{1}^{2\alpha_{1}}\geq P_{1}^{1/2-3\eta}$ and $\eta\in(0,1/12)$, we get

$\mathcal{M}_{1}\ll P_{1}^{-1/6+2\eta}\mathcal{P}_{f}(X)^{2}\leq
P_{1}^{-1/6+2\eta}\mathcal{P}_{f}(X)^{2}.$

Let now $2\leq j\leq J$, if $J\geq 2$. By definition, we have

$\mathcal{T}_{j}=\bigcup_{r\in\mathcal{I}_{j-1}}\mathcal{T}_{j,r},$

where $\mathcal{T}_{j,r}$ is the set of all $t\in\mathcal{T}_{j}$ such that
$|Q_{r,H_{j-1}}(1+it)|>e^{-\alpha_{j-1}r/H_{j-1}}$. Pointwise bounding
$|Q_{v,H_{j}}(1+it)|$ for each $v\in\mathcal{I}_{j}$ leads to

$\displaystyle\mathcal{M}_{j}$ $\displaystyle\leq H_{j}\log
Q_{j}\sum_{v\in\mathcal{I}_{j}}\sum_{r\in\mathcal{I}_{j-1}}e^{-2\alpha_{j}v/H_{j}}\int_{\mathcal{T}_{j}}|R_{v,H_{j}}(1+it)|^{2}dt$
$\displaystyle\leq(H_{j}\log
Q_{j})|\mathcal{I}_{j}||\mathcal{I}_{j-1}|e^{-2v_{0}\alpha_{j}+2\ell_{j}r_{0}\alpha_{j-1}}\int_{-X/h}^{X/h}|Q_{r_{0},H_{j-1}}(1+it)^{\ell}R_{v,H_{j}}(1+it)|^{2}dt,$

where $(r_{0},v_{0})\in\mathcal{I}_{j-1}\times\mathcal{I}_{j}$ yield the
maximal contribution among all such pairs, and
$\ell_{j}:=\lceil\frac{v/H_{j}}{r/H_{j-1}}\rceil$. Using Lemma 4.4, we get

$\mathcal{M}_{j}\ll_{B,C}(H_{j}\log
Q_{j})^{3}e^{-2v_{0}\alpha_{j}+2\ell_{j}r_{0}\alpha_{j-1}}\exp\left(2\ell\log(2B\ell)\right)\left(1+\frac{1}{h_{0}}\right)\mathcal{P}_{f}(X)^{2}.$

Minor modifications to the estimates in [26, Sec. 8.2] (with $h_{0}$ in place
of $h$ there), selecting $h_{0}$ sufficiently large in terms of $B$, shows
that

$\mathcal{M}_{j}\ll_{B,C}\frac{1}{j^{2}P_{1}}\mathcal{P}_{f}(X)^{2},$

whence it follows that

$\sum_{2\leq j\leq J}\mathcal{M}_{j}\ll P_{1}^{-1}\mathcal{P}_{f}(X)^{2}$

(the requirements of our parameters $P_{j},Q_{j}$ and $\alpha_{j}$ summarized
in Remark 4.6 are sufficient for this).
Finally, we consider $\mathcal{U}$. Set $H:=(\log X)^{\kappa}$, $P=\exp((\log
X)^{1-\kappa})$ and $Q=\exp(\log X/\log\log X)$, put
$\mathcal{I}:=[\left\lfloor H\log P\right\rfloor,H\log Q]\cap\mathbb{Z}$, and
define $Q_{v,H}$ and $R_{v,H}$ by

$Q_{v,H}(s):=\sum_{\begin{subarray}{c}P\leq p\leq Q\\\ e^{v/H}\leq p\leq
e^{(v+1)/H}\end{subarray}}f(p)p^{-s},\quad\quad
R_{v,H}(s):=\sum_{\frac{1}{3}Xe^{-v/H}<Xe^{-v/H}}\frac{f(m)}{m^{s}(1+\omega_{[P,Q]}(m))}.$

Combining Lemma 4.3 with Lemma 4.1 a) and the proof of Lemma 4.7 to control
those $n$ coprime to $P\leq p\leq Q$, we get that there is some
$v_{0}\in\mathcal{I}$ such that

$\displaystyle\int_{\mathcal{U}}|F(1+it)|^{2}dt$ $\displaystyle\ll_{B,C}(H\log
X)^{2}\int_{\mathcal{U}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}dt+\int_{-T}^{T}\left|\sum_{\begin{subarray}{c}X/3<n\leq
X\\\
(n,[P,Q])=1\end{subarray}}\frac{f(n)}{n^{1+it}}\right|^{2}dt+\mathcal{P}_{f}(X)^{2}\left(1+\frac{1}{h_{0}}\right)\left(\frac{1}{H}+\frac{1}{P}\right)$
$\displaystyle\ll_{B,C}(H\log
X)^{2}\int_{\mathcal{U}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}dt+\mathcal{P}_{f}(X)^{2}\left(\frac{1}{H}+\frac{1}{P}+\left(\frac{\log
P}{\log Q}\right)^{A}\right)$ (24) $\displaystyle\ll_{B,C}(H\log
X)^{2}\int_{\mathcal{U}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}dt+\left(\frac{\log\log
X}{(\log X)^{\kappa}}\right)^{\min\\{1,A\\}}\mathcal{P}_{f}(X)^{2},$

As in [26, Sec. 8.3] we may select a discrete subset
$\mathcal{V}\subset\mathcal{U}$ that is well-spaced, such that

$\int_{\mathcal{U}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}dt\ll\sum_{t\in\mathcal{V}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}.$

By assumption, for each $t\in\mathcal{V}$ we have
$|Q_{r_{0},H}(1+it)|>e^{-\alpha_{J}r_{0}/H_{J}}\geq P_{J}^{-\alpha_{J}}$, for
some $r_{0}\in\mathcal{I}_{J}$. We have $\log Q_{J+1}\geq\sqrt{\log X}$ by
definition, and (as mentioned in Remark 4.6) $\log
P_{J}\geq\frac{4B}{\eta}\log\log Q_{J+1}$, whence $(\log X)^{2B/\eta}\leq
P_{J}\leq Q_{J}\leq\exp(\sqrt{\log X})$. Applying Lemma 4.2(b) for each
$r_{0}\in\mathcal{I}_{J}$, we thus have

$|\mathcal{V}|\ll_{B}|\mathcal{I}_{J}|\exp\left(2\alpha_{J}(\log
P_{J})\left(1+\frac{\log X}{\log P_{J}}\right)+2B\frac{\log X\log\log X}{\log
P_{J}}\right)\leq X^{1/2-2\eta+o(1)}\cdot X^{\eta}=X^{1/2-\eta+o(1)}.$

We now split the set $\mathcal{V}$ into the subsets

$\displaystyle\mathcal{V}_{S}$
$\displaystyle:=\\{t\in\mathcal{V}:|Q_{v_{0},H}(1+it)|\leq(\log
X)^{-\frac{1}{2}B^{2}-10}\\}$ $\displaystyle\mathcal{V}_{L}$
$\displaystyle:=\\{t\in\mathcal{V}:|Q_{v_{0},H}(1+it)|>(\log
X)^{-\frac{1}{2}B^{2}-10}\\};$

the exponent $B^{2}/2$ is present in order to cancel the $\log X$ power that
arises from

$\mathcal{P}_{f^{2}}(X)\ll_{B}H(f;X)\mathcal{P}_{f}(X)^{2}\ll_{B}(\log
X)^{B^{2}}\mathcal{P}_{f}(X)^{2}.$

By a pointwise bound, Lemma 4.1(c) and the above estimate for
$|\mathcal{V}|\geq|\mathcal{V}_{S}|$, we obtain

$\displaystyle\sum_{t\in\mathcal{V}_{S}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}$
$\displaystyle\ll(\log
X)^{-B^{2}-19}(1+|\mathcal{V}_{S}|X^{-1/2})\frac{e^{v_{0}/H}}{X}\sum_{X/(3e^{v_{0}/H})<n\leq
X/e^{v_{0}/H}}|f(n)|^{2}$ $\displaystyle\ll_{B,C}(\log
X)^{-B^{2}-19}\mathcal{P}_{f^{2}}(X)\ll_{B}(\log
X)^{-19}\mathcal{P}_{f}(X)^{2}.$

Consider next the contribution from $\mathcal{V}_{L}$. Applying Lemma 4.2(b)
once again, this time using the condition $|Q_{v_{0},H}(1+it)|>(\log
X)^{-\frac{1}{2}B^{2}-10}$, to obtain

$|\mathcal{V}_{L}|\ll_{B}\exp\left((B^{2}+20)\left(\log\log
X\right)\left(1+\frac{\log X}{\log P}\right)+2B\frac{\log X\log\log X}{\log
P}\right)=\exp\left((\log X)^{\kappa+o_{B}(1)}\right).$

Recall that $\Delta=(2B+5)\kappa\in(0,1)$. Applying Lemma 3.5 (with
$Z=1/\delta=(\log X)^{\Delta}$) together with Lemma 4.2(a), noting that
$\kappa<1/3-\kappa$, we obtain

$\displaystyle\sum_{t\in\mathcal{V}_{L}}|Q_{v_{0},H}(1+it)R_{v_{0},H}(1+it)|^{2}$
$\displaystyle\ll_{A,B,C}\mathcal{P}_{f}(X)^{2}\left(\left(\frac{\log Q}{\log
P}\right)^{B}\delta^{1/2}+\left(\frac{\log Q}{\log
P}\right)^{3B}\frac{\log\log X}{(\log
X)^{\sigma}}\right)^{2}\sum_{t\in\mathcal{V}_{L}}|R_{v_{0},H}(1+it)|^{2}$
$\displaystyle\ll_{B}\frac{\mathcal{P}_{f}(X)^{2}}{(\log P)^{2}}\left((\log
X)^{2B\kappa-\Delta}+(\log
X)^{6B\kappa-2\sigma+o(1)}\right)\left(1+|\mathcal{V}_{L}|(\log
X)^{2}\exp\left(-(\log X)^{1/3-\kappa-o(1)}\right)\right)$
$\displaystyle\ll_{B}\mathcal{P}_{f}(X)^{2}\left((\log
X)^{2(B+1)\kappa-2-\Delta}+(\log X)^{(6B+2)\kappa-2-2\sigma+o(1)}\right).$

Combining this estimate with the one for $\mathcal{V}_{S}$, then plugging this
back into our estimate (4.5), we get

$\displaystyle\int_{\mathcal{U}}|F(1+it)|^{2}dt$
$\displaystyle\ll_{A,B,C}(\log X)^{2+2\kappa}\left((\log X)^{-19}+(\log
X)^{2(B+1)\kappa-2-\Delta}+(\log
X)^{(6B+2)\kappa-2-2\sigma+o(1)}\right)\mathcal{P}_{f}(X)^{2}$
$\displaystyle+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\mathcal{P}_{f}(X)^{2}$
$\displaystyle\ll\left((\log X)^{-15}+(\log X)^{2(B+2)\kappa-\Delta}+(\log
X)^{(6B+4)\kappa-2\sigma}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\mathcal{P}_{f}(X)^{2}$
$\displaystyle\ll\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\mathcal{P}_{f}(X)^{2},$

since $2(B+2)\kappa-\Delta=-\kappa$, and
$\sigma\geq\hat{\sigma}>(3B+5/2)\kappa$ by definition. This completes the
proof of Proposition 4.8.

###### Proof of Theorem 2.1.

Let $f\in\mathcal{M}(X;A,B,C;\gamma,\sigma)$. Set $h:=h_{1}/(\log h_{0})^{A}$,
and select $P_{1}=(\log h)^{40B/\eta}$ and $Q_{1}=h$. We may assume that $X$
is larger than any constant depending on $B$, since otherwise Theorem 2.1
follows with a sufficiently large implied constant; we may also assume $h$ is
larger than any constant depending on $B$, since otherwise the theorem follows
(again with a large enough implied constant depending at most on $B$) from
Remark 1.9 and Lemma 4.7 (taking $P=Q=3/2$, say).
By Lemma 4.7, we have

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\sum_{x-h_{1}<n\leq
x}f(m)-\frac{1}{h_{1}}\int_{n-h_{1}}^{n}u^{it_{0}}du\cdot\frac{1}{h_{2}}\sum_{\begin{subarray}{c}n-h_{2}<m\leq
n\end{subarray}}f(m)m^{-it_{0}}\right|^{2}$
$\displaystyle\ll_{A,B,C}\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\sum_{\begin{subarray}{c}n-h_{1}<m\leq
n\\\
m\in\mathcal{S}\end{subarray}}f(m)-\frac{1}{h_{1}}\int_{n-h_{1}}^{n}u^{it_{0}}du\cdot\frac{1}{h_{2}}\sum_{\begin{subarray}{c}n-h_{2}<m\leq
n\\\ m\in\mathcal{S}\end{subarray}}f(m)m^{-it_{0}}\right|^{2}$
$\displaystyle+\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{A}\mathcal{P}_{f}(X)^{2}.$

Set $\delta:=(\log X)^{-(2B+5)\kappa}$ once again, and note that
$t_{0}(fn^{-it_{0}},X)=0$ is admissible. Thus,

$\sum_{\begin{subarray}{c}X/3<n\leq X\\\
n\in\mathcal{S}\end{subarray}}\frac{f(n)n^{-it_{0}}}{n^{s}}=F(s+it_{0}).$

By the Cauchy-Schwarz inequality, we obtain

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\sum_{\begin{subarray}{c}x-h_{1}<m\leq
x\\\
m\in\mathcal{S}\end{subarray}}f(m)-\frac{1}{h_{1}}\int_{x-h_{1}}^{x}u^{it_{0}}du\cdot\frac{1}{h_{2}}\sum_{\begin{subarray}{c}x-h_{2}<m\leq
x\\\ m\in\mathcal{S}\end{subarray}}f(m)m^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll\frac{1}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\sum_{\begin{subarray}{c}x-h_{1}<m\leq
x\\\ m\in\mathcal{S}\end{subarray}}f(m)-\frac{1}{2\pi
h_{1}}\int_{I_{\delta}}F(1+it)\frac{x^{1+it}-(x-h_{1})^{1+it}}{1+it}dt\right|^{2}dx$
$\displaystyle+\frac{1}{X}\int_{X/2}^{X}\left|\frac{1}{h_{1}}\int_{x-h_{1}}^{x}u^{it_{0}}du\right|^{2}\left|\frac{1}{h_{2}}\sum_{\begin{subarray}{c}x-h_{2}<m\leq
x\\\ m\in\mathcal{S}\end{subarray}}f(m)m^{-it_{0}}-\frac{1}{2\pi
h_{2}}\int_{-\delta^{-1}}^{\delta^{-1}}F(1+it+it_{0})\frac{x^{1+it}-(x-h_{2})^{1+it}}{1+it}dt\right|^{2}dx$
$\displaystyle+\small\max_{X/2<x\leq
X}\left|\frac{1}{h_{1}}\int_{I_{\delta}}F(1+it)\frac{x^{1+it}-(x-h_{1})^{1+it}}{1+it}dt-\frac{1}{h_{1}}\int_{x-h_{1}}^{x}u^{it_{0}}du\cdot\frac{1}{h_{2}}\int_{-\delta^{-1}}^{\delta^{-1}}F(1+i(t+t_{0}))\frac{x^{1+it}-(x-h_{2})^{1+it}}{1+it}dt\right|^{2}\normalsize$
$\displaystyle=:\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}.$

Consider $\mathcal{I}_{3}$ first. Making a change of variables $w=t-t_{0}$ and
using $u^{iw}=x^{iw}+O(\delta^{-1}h_{1}/X)$ for $|w|\leq\delta^{-1}$ and
$u\in[x-h_{1},x]$, we see that

$\displaystyle\frac{1}{h_{1}}\int_{I_{\delta}}F(1+it)\frac{x^{1+it}-(x-h_{1})^{1+it}}{1+it}dt=\frac{1}{h_{1}}\int_{-\delta^{-1}}^{\delta^{-1}}F(1+i(t_{0}+w))\int_{x-h_{1}}^{x}u^{it_{0}+iw}dudw$
$\displaystyle=\frac{1}{h_{1}}\int_{x-h_{1}}^{x}u^{it_{0}}du\cdot\int_{-\delta^{-1}}^{\delta^{-1}}F(1+i(t_{0}+w))x^{iw}dw+O\left(\delta^{-2}\frac{h_{1}}{X}\max_{|w|\leq\delta^{-1}}|F(1+i(t_{0}+w))|\right).$

Similarly, we have

$\frac{1}{h_{2}}\int_{-\delta^{-1}}^{\delta^{-1}}F(1+i(t+t_{0}))\frac{x^{1+it}-(x-h_{2})^{1+it}}{1+it}dt=\int_{-\delta^{-1}}^{\delta^{-1}}F(1+i(t+t_{0}))x^{it}dt+O\left(\delta^{-2}\frac{h_{2}}{X}\max_{|t|\leq\delta^{-1}}|F(1+i(t+t_{0}))|\right).$

Applying partial summation, dropping the condition $n\in\mathcal{S}$, and then
using Lemma 3.2, we find

$\max_{|t|\leq\delta^{-1}}|F(1+i(t+t_{0}))|\ll\frac{1}{X}\sum_{X/3<n\leq
X}|f(n)|\ll_{B,C}\mathcal{P}_{f}(X),$

and thus as $h_{1}\leq h_{2}=X/(\log X)^{\hat{\sigma}/2}$ we obtain

$\mathcal{I}_{3}\ll_{B,C}\left(\delta^{-2}\frac{h_{2}}{X}\right)^{2}\mathcal{P}_{f}(X)^{2}\ll(\log
X)^{4(2B+5)\kappa-(8B+21)\kappa}\mathcal{P}_{f}(X)^{2}=(\log
X)^{-\kappa}\mathcal{P}_{f}(X).$

Next, we treat $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$. Applying Proposition
4.5 to each of these integrals and trivially bounding the $u$ integral in
$\mathcal{I}_{2}$, we find

$\displaystyle\mathcal{I}_{1}+\mathcal{I}_{2}$
$\displaystyle\ll\int_{[-X/h_{1},X/h_{1}]\backslash
I_{\delta}}|F(1+it)|^{2}dt+\max_{T\geq
X/h_{1}}\frac{X}{h_{1}T}\int_{T}^{2T}|F(1+it)|^{2}dt.$

By the argument in Remark 4.9 and our choice of $h$, the latter is bounded by

$\ll\int_{[-X/h,X/h]\backslash I_{\delta}}|F(1+it)|^{2}dt+(\log
h_{0})^{-A}\mathcal{P}_{f}(X)^{2}.$

But by Proposition 4.8 this integral is bounded by

$\ll_{A,B,C}\left(\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\mathcal{P}_{f}(X)^{2}\ll\left((\log
h)^{-30B}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\mathcal{P}_{f}(X)^{2}.$

Combining these steps, and using $A\leq B$ and $\log h\asymp\log h_{1}$, we
obtain

$\displaystyle\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h_{1}<n\leq
x}f(m)-\frac{1}{h_{1}}\int_{n-h_{1}}^{n}u^{it_{0}}du\cdot\frac{1}{h_{2}}\sum_{n-h_{2}<m\leq
n}f(m)m^{-it_{0}}\right|^{2}dx$
$\displaystyle\ll_{A,B,C}\left(\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{A}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\min\\{1,A\\}}\right)\mathcal{P}_{f}(X)^{2},$

which completes the proof. ∎

###### Proof of Corollary 1.8.

It suffices to show that
$\mathcal{M}(X;A,B,C)\subseteq\mathcal{M}(X;A,B,C;1,\sigma)$ for any
$0<\sigma<\sigma_{A,B}$, where $\sigma_{A,B}$ is given by (9), after which
point the result follows from Theorem 2.1.
Suppose $f\in\mathcal{M}(X;A,B,C)$. By definition, $f$ is $(A,X)$ non-
vanishing in the sense of [24], and so satisfies the condition (6) (or
equivalently, $\gamma=1$ is admissible in (7)). Observe furthermore that upon
defining $\tilde{f}_{B}:=f\cdot B^{-\Omega}$, $\tilde{f}_{B}$ takes values in
$\mathbb{U}$ and satisfies

$\sum_{z<p\leq w}\frac{|\tilde{f}_{B}(p)|}{p}\geq\frac{A}{B}\sum_{z<p\leq
w}\frac{1}{p}-O_{A,B}\left(\frac{1}{\log z}\right)\text{ for all }2\leq z\leq
w\leq X,$

with $A/B\in(0,1]$ necessarily. By [24, Lem. 5.1(i)], we get that

$\displaystyle\rho(f,n^{it};X)^{2}$ $\displaystyle=B\sum_{p\leq
X}\frac{|\tilde{f}_{B}(p)|-\text{Re}(\tilde{f}_{B}(p)p^{-it})}{p}$ (25)
$\displaystyle\geq B\rho\min\\{\log\log X,3\log(|t-t_{0}|\log
X+1)\\}+O_{\rho,A}(1),$

for any $0<\rho<\rho_{A/B}$ with $\rho_{\alpha}$ defined (see [24, (14)]) by

$\rho_{\alpha}=\frac{\alpha}{3}\left(1-\text{sinc}(\pi\alpha/2)\right)>0.$

Since $\sigma_{A,B}=B\rho_{A/B}$, this implies that condition (8) holds with
any $0<\sigma<\sigma_{A,B}$. This completes the proof. ∎

##  5\. Applications

###  5.1. Proof of Theorem 1.4

Let $f$ be a primitive, Hecke-normalized holomorphic cusp form without complex
multiplication of fixed even weight $k\geq 2$ and level 1. Write the Fourier
expansion of $f$ at $\infty$ as

$f(z)=\sum_{n\geq
1}\lambda_{f}(n)n^{\frac{k-1}{2}}e(nz),\quad\quad\text{Im}(z)>0,$

where $\\{\lambda_{f}(n)\\}_{n}$ is the sequence of normalized Fourier
coefficients with $\lambda_{f}(1)=1$. As noted in Section 1.3.2 $\lambda_{f}$
is a multiplicative function. Deligne’s proof of the Ramanujan conjecture for
$f$ shows that $|\lambda_{f}(p)|\leq 2$ and $|\lambda_{f}(n)|\leq d(n)$ for
all primes $p$ and positive integers $n$. Moreover, the quantiative Sato-Tate
theorem of Thorner [44], which is based on the deep results of Newton and
Thorne [30], shows that for any $[a,b]\subseteq[-2,2]$,

(26) $|\\{p\leq
X:\lambda_{f}(p)\in[a,b]\\}|=\left(\frac{1}{\pi}\int_{a}^{b}\sqrt{1-(v/2)^{2}}dv\right)\int_{2}^{X}\frac{dt}{\log
t}+O\left(\frac{X\log(k\log X)}{(\log X)^{3/2}}\right).$

Recall that

$c_{\alpha}=\frac{2^{\alpha}}{\sqrt{\pi}}\frac{\Gamma\left(\frac{\alpha+1}{2}\right)}{\Gamma(\alpha/2+2)}.$

Using this data, we will prove the following.

######  Proposition 5.1.

Let $\alpha>0$. There is a constant $\delta=\delta(\alpha)>0$ such that
$|\lambda_{f}|^{\alpha}\in\mathcal{M}(X;c_{\alpha},2^{\alpha},2;1/2-\varepsilon,\delta)$.

Using Proposition 5.1 we will be able to apply Theorem 1.7 in order to derive
Corollary 1.1.
We will check that $|\lambda_{f}|^{\alpha}$ satisfies the required hypotheses
in the following lemmas.

######  Lemma 5.2.

For any $2\leq z<w$ we have

$\sum_{z<p\leq w}\frac{|\lambda_{f}(p)|^{\alpha}}{p}=c_{\alpha}\sum_{z<p\leq
w}\frac{1}{p}+O((\log z)^{-1/2+o(1)}).$

Similarly, we have

$\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{2\alpha}}{p}=c_{2\alpha}\log\log
X+O(1).$

###### Proof.

Let $\beta\in\\{\alpha,2\alpha\\}$. By partial summation,

$\sum_{z<p\leq
w}\frac{|\lambda_{f}(p)|^{\beta}}{p}=\int_{0}^{2}u^{\beta}d\\{\sum_{\begin{subarray}{c}z<p\leq
w\\\ |\lambda_{f}(p)|\leq
u\end{subarray}}\frac{1}{p}\\}=2^{\beta}\sum_{z<p\leq
w}\frac{1}{p}-\beta\int_{0}^{2}\left(\sum_{\begin{subarray}{c}z<p\leq w\\\
|\lambda_{f}(p)|\leq u\end{subarray}}\frac{1}{p}\right)u^{\beta-1}du.$

Fix $u\in(0,2]$, and let $I_{u}:=[0,u]\cup[-u,0]$ so that
$|\lambda_{f}(p)|\leq u$ if and only if $\lambda_{f}(p)\in I_{u}$. By partial
summation and (26),

$\displaystyle\sum_{\begin{subarray}{c}z<p\leq w\\\ \lambda_{f}(p)\in
I_{u}\end{subarray}}\frac{1}{p}$
$\displaystyle=\frac{2}{\pi}\int_{0}^{u}\sqrt{1-(v/2)^{2}}dv\cdot\int_{z}^{w}\frac{dy}{y\log
y}+O\left(\frac{\log\log z}{(\log z)^{3/2}}+\int_{z}^{w}\frac{\log\log
y\,dy}{y(\log y)^{3/2}}\right)$
$\displaystyle=\left(\frac{2}{\pi}\int_{0}^{u}\sqrt{1-(v/2)^{2}}dv\right)\log(\log
w/\log z)+O((\log z)^{-1/2+o(1)}).$

Multiplying the main term by $\beta u^{\beta-1}$ and integrating in $u$, we
obtain $I_{\beta}\log(\log w/\log z)+O(1/\log z)$, where

$\displaystyle I_{\beta}$
$\displaystyle:=\frac{2\beta}{\pi}\int_{0}^{2}u^{\beta-1}\int_{0}^{u}\sqrt{1-(v/2)^{2}}dvdu=\frac{2\beta}{\pi}\int_{0}^{2}\left(\int_{v}^{2}u^{\beta-1}du\right)\sqrt{1-(v/2)^{2}}dv$
$\displaystyle=\frac{2^{\beta+1}}{\pi}\int_{0}^{2}(1-(v/2)^{\beta})\sqrt{1-(v/2)^{2}}dv=2^{\beta}-\frac{2^{\beta+1}}{\pi}\int_{0}^{2}(v/2)^{\beta}\sqrt{1-(v/2)^{2}}dv.$

Making the change of variables $t:=(v/2)^{2}$, we find that

$2^{\beta}-I_{\beta}=\frac{2^{\beta+1}}{\pi}\int_{0}^{1}t^{(\beta-1)/2}(1-t)^{1/2}dt=\frac{2^{\beta+1}}{\pi}\frac{\Gamma\left(\frac{\beta+1}{2}\right)\Gamma(3/2)}{\Gamma(\beta/2+2)}=\frac{2^{\beta}}{\sqrt{\pi}}\frac{\Gamma\left(\frac{\beta+1}{2}\right)}{\Gamma(\beta/2+2)}=c_{\beta}.$

It follows, therefore, that

$\displaystyle\sum_{z<p\leq w}\frac{|\lambda_{f}(p)|^{\alpha}}{p}$
$\displaystyle=I_{\alpha}\sum_{z<p\leq w}\frac{1}{p}+O((\log
z)^{-1/2+o(1)})=c_{\alpha}\sum_{z<p\leq w}\frac{1}{p}+O((\log
z)^{-1/2+o(1)}),$ $\displaystyle\sum_{z<p\leq
w}\frac{|\lambda_{f}(p)|^{2\alpha}}{p}$
$\displaystyle=I_{2\alpha}\sum_{z<p\leq w}\frac{1}{p}+O((\log
z)^{-1/2+o(1)})=c_{2\alpha}\sum_{z<p\leq w}\frac{1}{p}+O((\log
z)^{-1/2+o(1)}),$

and both claims follow. ∎

To obtain uniform lower bound estimates for
$\rho(|\lambda_{f}|^{\alpha},n^{it};X)^{2}$ we will need some control over the
product $|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log p))$, on average over $p$. In
some ranges of $t$ this is furnished by the following lemma.

######  Lemma 5.3.

Let $|t|\geq 1$. There is a constant $c=c(\alpha)>0$ such that if
$Y\geq(|t|+3)^{2}$ and $Y$ is sufficiently large then

$\sum_{Y<p\leq 2Y}|\lambda_{f}(p)|^{\alpha}|1-p^{it}|^{2}\geq c\frac{Y}{\log
Y}.$

###### Proof.

We adapt an argument due to Goldfeld and Li [4, Lem. 12.12 and 12.15] and
Humphries [16, Lem. 2.1]. Let $\eta\in(0,1/2)$ be a parameter to be chosen
later. By the prime number theorem for Rankin-Selberg $L$-functions [31, Thm.
2] we have

$\sum_{Y<p\leq 2Y}|\lambda_{f}(p)|^{2}\log p=(1+o(1))Y.$

Since $|\lambda_{f}(p)|^{2}\leq 4$ for all $p$, invoking the usual prime
number theorem we obtain

$\displaystyle(1+o(1))Y$
$\displaystyle\leq\eta^{2}\sum_{\begin{subarray}{c}Y<p\leq 2Y\\\
|\lambda_{f}(p)|\leq\eta\end{subarray}}\log p+4(\log 2Y)|\\{Y<p\leq
2Y:|\lambda_{f}(p)|>\eta\\}|$ $\displaystyle\leq(\eta^{2}+o(1))Y+4(\log
2Y)|\\{Y<p\leq 2Y:|\lambda_{f}(p)|>\eta\\}|,$

which rearranges as

(27) $|\\{Y<p\leq
2Y:|\lambda_{f}(p)|>\eta\\}|\geq\left(\frac{1-\eta^{2}}{4}-o(1)\right)\frac{Y}{\log
Y}.$

Next, we estimate the cardinality

$\displaystyle|\\{Y<p\leq 2Y:|1-p^{it}|\leq\eta\\}|$
$\displaystyle=|\\{Y<p\leq 2Y:|\sin((t\log p)/2)|\leq\eta/2\\}|.$

Set $\beta:=\sin^{-1}(\eta/2)/\pi\in[0,1/2]$, Whenever $\sin(t\log
p/2)\in[-\eta/2,\eta/2]$ there is $m\in\mathbb{Z}$ such that $(t\log
p)/2\in[\pi(m-\beta),\pi(m+\beta)]$. By Jordan’s inequality,
$\beta\leq\frac{1}{2}\sin(\pi\beta)=\frac{\eta}{4}$, and we see that

$\displaystyle|\\{Y<p\leq 2Y:|1-p^{it}|\leq\eta\\}|\leq|\\{Y<p\leq 2Y:\|(t\log
p)/(2\pi)\|\leq\eta/4\\}|,$

where $\|t\|:=\min_{m\in\mathbb{Z}}|t-m|$. Splitting up the primes $Y<p\leq
2Y$ according to the nearest integer $m$ to $(t\log p)/(2\pi)$, the latter may
be bounded above as

$\leq\sum_{\frac{|t|\log Y+\eta/4}{2\pi}\leq
m\leq\frac{|t|\log(2Y)-\eta/4}{2\pi}}\left(\pi\left(e^{\frac{2\pi
m+\eta/4}{|t|}}\right)-\pi\left(e^{\frac{2\pi m-\eta/4}{|t|}}\right)\right).$

For each $m$ we have $e^{\frac{2\pi
m+\eta/4}{|t|}}\leq\left(1+\frac{\eta}{2|t|}\right)e^{\frac{2\pi
m-\eta/4}{|t|}}$. By the Brun-Titchmarsh theorem we get, uniformly over all
$m$ in the sum,

$\pi\left(e^{\frac{2\pi m+\eta/4}{|t|}}\right)-\pi\left(e^{\frac{2\pi
m-\eta/4}{|t|}}\right)\leq\frac{\eta}{|t|}\frac{e^{2\pi(m-\eta/4)/|t|}}{2\pi(m-\eta/4)/|t|-\log(|t|/\eta)}\leq\eta\frac{2Y}{|t|(\log
Y-(\log Y)/2)}\leq 8\eta\frac{Y}{|t|\log Y},$

for $Y$ large enough. Since there are $\leq 1+|t|(\log(2Y)-\log Y)/(2\pi)\leq
2|t|$ integers in the range of summation, we obtain

(28) $|\\{Y<p\leq 2Y:|1-p^{it}|\leq\eta\\}|\leq 16\eta\frac{Y}{\log Y}.$

We deduce from (27) and (28) that

$\displaystyle|\\{Y<p\leq 2Y:|\lambda_{f}(p)|>\eta\text{ and
}|1-p^{it}|>\eta\\}|$ $\displaystyle\geq|\\{Y<p\leq
2Y:|\lambda_{f}(p)|>\eta\\}|-|\\{Y<p\leq 2Y:|1-p^{it}|\leq\eta\\}|$
$\displaystyle\geq\max\left\\{0,\frac{1-64\eta-\eta^{2}}{4}-o(1)\right\\}\cdot\frac{Y}{\log
Y}.$

Selecting $\eta=\frac{1}{128}$, we get that if $Y$ is sufficiently large,

$|\\{Y<p\leq 2Y:|\lambda_{f}(p)|>\eta\text{ and
}|1-p^{it}|>\eta\\}|\geq\frac{Y}{10\log Y}.$

Finally, we obtain that

$\sum_{Y<p\leq
2Y}|\lambda_{f}(p)|^{\alpha}|1-p^{it}|^{2}>\eta^{2+\alpha}|\\{Y<p\leq
2Y:|\lambda_{f}(p)|>\eta\text{ and
}|1-p^{it}|>\eta\\}|\geq\frac{\eta^{2+\alpha}}{10}\cdot\frac{Y}{\log Y},$

which proves the claim with $c:=2^{-10(2+\alpha)}$. ∎

We now obtain lower bounds for $\rho(|\lambda_{f}|^{\alpha},n^{it};X)^{2}$ for
all $|t|\leq X$.

######  Lemma 5.4.

There is a $\delta=\delta(\alpha)>0$ such that whenever $|t|\leq X$ we have

$\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p))}{p}\geq\delta\min\\{\log\log X,\log(1+|t|\log X)\\}-O(1).$

Moreover, we may select $t_{0}(|\lambda_{f}|^{2},X)=0$.

###### Proof.

We may assume that $X$ is sufficiently large, else the estimate given is
trivial. When $|t|\leq 1/\log X$ the claim is vacuous. Thus, we may focus on
the case $1/\log X<|t|\leq X$. We consider the ranges $1/\log X<|t|\leq 1$,
$1<|t|\leq\log X$ and $\log X<|t|\leq X$ separately. Throughout we will
introduce an auxiliary parameter $2\leq Y\leq X$, chosen case by case, and use
the inequality

$\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p)}{p}\geq\sum_{Y\leq p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p))}{p}.$

Suppose first that $1/\log X<|t|\leq 1$. Let $Y:=e^{1/|t|}$ and write

$R(X):=\sum_{p\leq X}|\lambda_{f}(p)|^{\alpha}-c_{\alpha}\pi(X),$

so that by arguments analogous to those of Lemma 5.2, $R(X)\ll X/(\log
X)^{3/2-o(1)}$. By partial summation and Lemma 5.2,

$\displaystyle\sum_{Y\leq p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}\cos(t\log
p)}{p}$ $\displaystyle=c_{\alpha}\int_{Y}^{X}\frac{\cos(t\log
u)}{u}\frac{du}{\log u}+O\left(\frac{1}{\log
Y}\right)-\int_{Y}^{X}R(u)\left(\cos(t\log u)-t\sin(t\log
u)\right)\frac{du}{u^{2}}$ $\displaystyle=c_{\alpha}\int_{1}^{|t|\log
X}\cos(tv/|t|)\frac{dv}{v}+O\left((1+|t|)(\log Y)^{-1/2+o(1)}\right)\ll 1,$

the bound in the last step arising from setting $v:=|t|\log u$ and integrating
by parts. Thus, in light of Lemma 5.2

$\sum_{Y<p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p))}{p}=c_{\alpha}\log(1+|t|\log X)-O(1).$

Next, we consider the intermediate range $1<|t|\leq\log X$. Here, we set
$Y:=(10\log X)^{2}$, employ a dyadic decomposition and apply Lemma 5.3 to
obtain, when $X$ is large enough,

$\displaystyle\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p))}{p}$ $\displaystyle\geq\frac{1}{2}\sum_{Y<2^{j}\leq
X/2}2^{-(j+1)}\sum_{2^{j}<p\leq
2^{j+1}}|\lambda_{f}(p)|^{\alpha}|1-p^{it}|^{2}\geq\frac{c}{4\log
2}\sum_{Y<2^{j}\leq X/2}\frac{1}{j}$ $\displaystyle=\frac{c}{4\log
2}\log\left(\frac{\log X}{\log Y}\right)-O(1)\geq\frac{c}{8\log 2}\log\log
X-O(1),$

for some $c=c(\alpha)>0$.
Finally, assume that $\log X\leq|t|\leq X$. In this case, put
$Y:=\exp\left((\log X)^{2/3+\varepsilon}\right)$, where $\varepsilon>0$ is
small. Let $m\geq 1$ be an integer parameter to be chosen later. By Hölder’s
inequality,

$\displaystyle\left|\sum_{Y<p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}\cos(t\log
p)}{p}\right|\leq\left(\sum_{Y<p\leq X}\frac{|\lambda_{f}(p)|^{\frac{2\alpha
m}{2m-1}}}{p}\right)^{1-1/2m}\cdot\left(\sum_{Y<p\leq X}\frac{\cos(t\log
p)^{2m}}{p}\right)^{1/2m}.$

We can bound $|\lambda_{f}(p)|^{2\alpha
m/(2m-1)}\leq|\lambda_{f}(p)|^{\alpha}2^{\alpha/(2m-1)}$, so that by Lemma 5.2
the first sum is

$\leq 2^{\alpha/(2m)}c_{\alpha}^{1-1/(2m)}(\log(\log X/\log
Y))^{1-1/2m}+O_{m}(1).$

Now, we can write

$\cos(t\log
p)^{2m}=2^{-2m}(p^{it}+p^{-it})^{2m}=2^{-2m}\binom{2m}{m}+2^{-2m}\sum_{0\leq
j\leq m-1}\binom{2m}{j}(p^{2i(m-j)t}+p^{-2i(m-j)t}).$

By the zero-free region of the Riemann zeta function (see e.g., [25, Lem. 2])
we have

$\max_{1\leq|l|\leq m}\left|\sum_{Y<p\leq
X}\frac{1}{p^{1+ilt}}\right|\ll\frac{\log X}{1+|t|}+(\log X)^{-10}\ll 1.$

It follows that

$\sum_{Y<p\leq X}\frac{\cos(t\log p)^{2m}}{p}=2^{-2m}\binom{2m}{m}\log(\log
X/\log Y)+O_{m}(1),$

and therefore in sum we have

$\left|\sum_{Y<p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}\cos(t\log
p)}{p}\right|\leq
c_{\alpha}\left(\frac{2^{\alpha}}{c_{\alpha}}\right)^{1/(2m)}\left(2^{-2m}\binom{2m}{m}\right)^{1/2m}\log\left(\frac{\log
X}{\log Y}\right)+O_{m}(1).$

Using the bounds $\sqrt{2\pi n}(n/e)^{n}\leq n!\leq 2\sqrt{2\pi n}(n/e)^{n}$,
valid for all $n\in\mathbb{N}$ (see e.g., [32]), we get that

$\left(\frac{2^{\alpha}}{c_{\alpha}}\right)^{1/(2m)}\left(2^{-2m}\binom{2m}{m}\right)^{1/(2m)}\leq\left(\frac{2^{\alpha+1}}{c_{\alpha}\sqrt{\pi
m}}\right)^{1/(2m)},$

and thus taking $m\geq m_{0}(\alpha)$, we obtain

$\left|\sum_{Y<p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}\cos(t\log
p)}{p}\right|\leq 2^{-1/(2m)}c_{\alpha}\log(\log X/\log Y)+O(1),$

for $\eta\in(0,1)$. We thus obtain in this case that

$\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log p))}{p}\geq
c_{\alpha}(1-2^{-1/(2m)})\left(\frac{1}{3}-\varepsilon\right)\log\log X-O(1).$

Combining our estimates from each of these ranges and putting
$\delta:=c_{\alpha}\min\\{c/(8\log 2),\frac{1}{4}(1-2^{-1/(2m)})\\}$, we
deduce that

$\sum_{p\leq X}\frac{|\lambda_{f}(p)|^{\alpha}(1-\cos(t\log
p))}{p}\geq\delta\min\\{\log\log X,\log(1+|t|\log X)\\}-O(1).$

This completes the proof of the lower bound.
Finally, note that $\rho(|\lambda_{f}|^{\alpha},1;X)=0$, so $t_{0}=0$ is a
minimizer of $t\mapsto\rho(|\lambda_{f}|^{\alpha},n^{it};X)^{2}$. This
completes the proof. ∎

###### Proof of Proposition 5.1.

Let $X$ be large. By Deligne’s theorem we have $|\lambda_{f}(p)|^{\alpha}\leq
2^{\alpha}$ for all $p$. In addition, we have $|\lambda_{f}(n)|^{\alpha}\leq
d(n)^{\alpha}\leq d_{2^{\alpha+1}}(n)^{\max\\{1,\alpha\\}}$ for all $n$. By
Lemma 5.2 we see that we can take $A=c_{\alpha}$ and any $\gamma\in(0,1/2)$ in
an estimate of the form

$\sum_{z<p\leq w}\frac{|\lambda_{f}(p)|^{\alpha}}{p}\geq
c_{\alpha}\sum_{z<p\leq w}\frac{1}{p}-O\left(\frac{1}{(\log
z)^{\gamma}}\right)\text{ for }2\leq z\leq w\leq X,$

and finally by Lemma 5.4 there is a constant $\delta>0$ such that

$\sum_{p\leq
X}\frac{|\lambda_{f}(p)|^{\alpha}-\text{Re}(|\lambda_{f}(p)|^{\alpha}p^{-it})}{p}\geq\delta\min\\{\log\log
X,\log(1+|t-t_{0}|\log X)\\}-O(1),$

as $t_{0}=0$ is admissible. Thus, by definition,
$|\lambda_{f}|^{\alpha}\in\mathcal{M}(X;c_{\alpha},2^{\alpha},\max\\{1,\alpha\\};1/2-\varepsilon,\delta)$
for any $\varepsilon\in(0,1/2)$ and $X$ large, as claimed. ∎

###### Proof of Theorem 1.4.

In view of Proposition 5.1 we may apply Theorem 1.7 to the function
$|\lambda_{f}|^{\alpha}$. We see that if $h=h_{0}H(|\lambda_{f}|^{\alpha};X)$
and $h_{0}\rightarrow\infty$ then

$\displaystyle\frac{1}{X}\int_{X}^{2X}\left|\frac{1}{h}\sum_{x<n\leq
x+h}|\lambda_{f}(n)|^{\alpha}-\frac{1}{X}\sum_{X<n\leq
2X}|\lambda_{f}(n)|^{\alpha}\right|^{2}dx$
$\displaystyle\ll\left(\left(\frac{\log\log h_{0}}{\log
h_{0}}\right)^{c_{\alpha}}+\frac{\log\log X}{(\log X)^{\theta}}\right)(\log
X)^{2(c_{\alpha}-1)},$

where $\theta=\theta(\alpha)>0$. Furthermore, by Lemma 5.2, we have

$H(|\lambda_{f}|^{\alpha};X)\asymp\exp\left(\sum_{p\leq
X}\frac{|\lambda_{f}(p)|^{2\alpha}-2|\lambda_{f}(p)|^{\alpha}+1}{p}\right)\asymp
d_{\alpha}\log X,$

where $d_{\alpha}=c_{2\alpha}-2c_{\alpha}+1$. Thus, changing $h_{0}$ by a
constant factor, we deduce that $h=h_{0}(\log X)^{d_{\alpha}}$ can be taken in
the above estimate. The claim follows. ∎

###### Proof of Corollary 1.1.

When $\alpha=2$ we have $c_{2}=1$ and $d_{2}=c_{4}-2c_{2}+1=1$. Hence,
$H(|\lambda_{f}|^{2};X)\asymp\log X$. Since

$c_{f}=\frac{1}{X}\sum_{X<n\leq 2X}|\lambda_{f}(n)|^{2}+O(X^{-2/5}),$

the claim for $|\lambda_{f}|^{2}$ follows immediately from Theorem 1.4.
It remains to consider $g_{f}(n)=\sum_{d^{2}|n}|\lambda_{f}(n/d^{2})|^{2}$.
Clearly, $g_{f}(n)\leq\sum_{e|n}d(n/e)^{2}\leq d(n)^{3}$, and
$g_{f}(p)=|\lambda_{f}(p)|^{2}$ for all primes $p$. By Proposition 5.1 we have
$g_{f}\in\mathcal{M}(X;1,4,3;1/2-\varepsilon,\delta)$ for any
$\varepsilon\in(0,1/2)$, and $H(g_{f};X)=H(|\lambda_{f}|^{2};X)$. The result
now follows in the same was as for $|\lambda_{f}|^{2}$, using

$d_{f}=\frac{1}{X}\sum_{X<n\leq 2X}g_{f}(n)+O(X^{-2/5})$

in this case. ∎

###  5.2. Proof of Theorem 1.2

Fix $m\geq 2$, let $\mathbb{A}$ denote the adeles over $\mathbb{Q}$, and let
$\pi$ be a fixed cuspidal automorphic representation for $GL_{m}(\mathbb{A})$
with unitary central character normalized so that it is trivial on the
diagonally embedded copy of $\mathbb{R}^{+}$. We write $q_{\pi}$ to denote the
conductor of $\pi$. We assume that $\pi$ satisfies GRC, and we write
$\lambda_{\pi}(n)$ to denote the $n$th coefficient of the standard
$L$-function of $\pi$. The key result of this subsection is the following
analogue of Proposition 5.1.

######  Proposition 5.5.

With the above notation, we have
$|\lambda_{\pi}|^{2}\in\mathcal{M}(X;1,m^{2},2)$ whenever $X$ is sufficiently
large in terms of $q(\pi)$, and $t_{0}(|\lambda_{\pi}|^{2},X)=0$ is
admissible.

For primes $p\nmid q_{\pi}$ we have
$|\lambda_{\pi}(p)|^{2}=\lambda_{\pi\otimes\tilde{\pi}}(p)$, where
$\tilde{\pi}$ denotes the contragredient representation of $\pi$ and
$\pi\otimes\tilde{\pi}$ is the Rankin-Selberg convolution of $\pi$ and
$\tilde{\pi}$. As $\pi$ is fixed, the primes $p|q_{\pi}$ will cause no harm to
our estimates.

######  Lemma 5.6.

There is a constant $c=c(m)>0$ such that

$\sum_{p\leq X}|\lambda_{\pi}(p)|^{2}\log p=X+O_{\pi}(Xe^{-c\sqrt{\log
X}})\text{ as $X\rightarrow\infty$.}$

In particular, we have $\sum_{z<p\leq
w}\frac{|\lambda_{\pi}(p)|^{2}}{p}=\sum_{z<p\leq
w}\frac{1}{p}+O_{\pi}(\frac{1}{\log z})$ for any $z<p\leq w$.

###### Proof.

Assume $X$ is sufficiently large relative to $q_{\pi}$. Set
$f:=\pi\otimes\tilde{\pi}$, and write $\Lambda_{f}(n)$ to denote the $n$th
coefficient of the logarithmic derivative $-\frac{L^{\prime}}{L}(s,f)$.
Combining [18, Thm. 5.13] (see the remarks that follow the statement for a
discussion relevant to the case of a Rankin-Selberg convolution) with [18,
Exer. 6], we deduce that

$\sum_{p\leq X}|\lambda_{\pi}(p)|^{2}\log p=\sum_{n\leq
X}\Lambda_{f}(n)+O_{\pi}(\sqrt{X}\log^{2}X)=X+O_{\pi}\left(X\exp\left(-c^{\prime}m^{-4}\sqrt{\log
X}\right)\right),$

for some absolute constant $c^{\prime}>0$ (note that the exceptional zero
plays no role when $X$ is large enough). This implies the first claim. The
second follows immediately from the first statement by partial summation. ∎

###### Proof of Proposition 5.5.

Lemma 5.6 implies that (6) holds with $A=1$, and by GRC we have
$|\lambda_{\pi}(n)|^{2}\leq d_{m}(n)^{2}$. It thus follows that
$|\lambda_{\pi}|^{2}\in\mathcal{M}(X;1,m^{2},2)$, as claimed. That the choice
$t_{0}(|\lambda_{\pi}|^{2},X)=0$ is admissible is obvious, as in the proof of
Proposition 5.1. ∎

###### Proof of Theorem 1.2.

Recall that $m$ and $\pi$ are fixed. A direct application of Corollary 1.8
shows (after replacing $X$ by $2X$ an $x-h$ by $x$) that there is
$\kappa=\kappa(m)>0$ such that

$\displaystyle\frac{1}{X}\int_{X}^{2X}\left|\frac{1}{h}\sum_{x<n\leq
x+h}|\lambda_{\pi}(n)|^{2}-\frac{1}{X}\sum_{X<n\leq
2X}|\lambda_{\pi}(n)|^{2}\right|^{2}dx\ll_{m}\left(\frac{\log\log h_{0}}{\log
h_{0}}+\frac{\log\log X}{(\log
X)^{\kappa}}\right)\mathcal{P}_{|\lambda_{\pi}|^{2}}(X)^{2},$

where $h=h_{0}H(|\lambda_{\pi}|^{2};X)$ and $10\leq h_{0}\leq
X/(10H(|\lambda_{\pi}|^{2};X))$. By Lemma 5.6
$\mathcal{P}_{|\lambda_{\pi}|^{2}}(X)\ll_{m}1$. We also have

$H(|\lambda_{\pi}|^{2};X)\asymp_{m}\exp\left(\sum_{p\leq
X}\frac{|\lambda_{\pi}(p)|^{4}-2|\lambda_{\pi}(p)|^{2}+1}{p}\right)\ll\exp\left(\sum_{p\leq
X}\frac{m^{2}|\lambda_{\pi}(p)|^{2}-1}{p}\right)\ll(\log X)^{m^{2}-1}.$

It follows that our variance estimate holds if $h\geq h_{0}(\log X)^{m^{2}-1}$
and $10\leq h_{0}\leq X/(10(\log X)^{m^{2}-1})$, and the proof of the theorem
is complete. ∎

###  5.3. Proof of Corollary 1.6

Recall that

$\Delta(n):=\max_{u\in\mathbb{R}}\sum_{\begin{subarray}{c}d|n\\\ e^{u}\leq
d<e^{u+1}\end{subarray}}1.$

Given $\theta\in\mathbb{R}$, write also

$d(n,\theta):=\sum_{d|n}d^{i\theta}.$

This is clearly a multiplicative function, with $d(n)=d(n,0)$. In
probabilistic terms, it is also $d(n)$ times the characteristic function of
the distribution function

$\mathcal{D}_{n}(v):=\frac{1}{d(n)}\sum_{\begin{subarray}{c}d|n\\\ d\leq
e^{v}\end{subarray}}1,\quad v\in\mathbb{R}.$

introduced in Section 1.3. The following general bounds for concentration
functions in terms of characteristic functions allow us to relate $\Delta(n)$
with integral averages of $d(n,\theta)$.

######  Lemma 5.7.

There are constants $c_{2}>c_{1}>0$ such that, uniformly in $n\in\mathbb{N}$,

$c_{1}\frac{1}{d(n)}\int_{0}^{1}|d(n,\theta)|^{2}d\theta\leq\Delta(n)\leq
c_{2}\int_{0}^{1}|d(n,\theta)|d\theta.$

###### Proof.

This is a special case of [12, Lem. 30.2]. ∎

By Lemma 5.7 we find that for any $x\in[X/2,X]$ and $10\leq h\leq X$,

(29) $\sum_{x-h<n\leq x}\Delta(n)\gg\int_{0}^{1}\sum_{x-h<n\leq
x}\frac{|d(n,\theta)|^{2}}{d(n)}d\theta.$

For $\theta\in\mathbb{R}$ and $n\in\mathbb{N}$ we write
$f_{\theta}(n):=|d(n,\theta)|^{2}/d(n)$.

######  Corollary 5.8.

Let $\theta\in(1/\log X,1]$. Let $10\leq h_{0}\leq X/\log X$ and put
$h:=h_{0}(\theta^{-1}\log X)^{1/2}$. Then there is a constant $\kappa_{1,2}>0$
such that for any $0<\kappa<\kappa_{1,2}$,

$\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)-\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right|^{2}dx\ll\left(\frac{\log\log h_{0}}{\log
h_{0}}+\frac{\log\log X}{(\log X)^{\kappa}}\right),$

The implicit constant is independent of $\theta$.

To this end, we prove that $f_{\theta}\in\mathcal{M}(X;1,2,1)$ for all $1/\log
X<\theta\leq 1$, which is the purpose of the following lemmas.

######  Lemma 5.9.

Let $\theta\in(0,1]$, and let $\beta:=\min\\{1/\theta,\log X\\}$. Then

$\sum_{p\leq X}\frac{f_{\theta}(p)}{p}=\log(\beta\log X)+O(1).$

Similarly,

$H(f_{\theta};X)\asymp(\beta\log X)^{1/2}.$

###### Proof.

Observe that for each $p$,

$f_{\theta}(p)=\frac{1}{2}|1+p^{i\theta}|^{2}=1+\cos(\theta\log p).$

Put $Y:=\min\\{X,\exp(1/\theta)\\}=e^{\beta}$. For $p\leq Y$ we have
$\cos(\theta\log p)=1+O(\theta^{2}(\log p)^{2})$, so that by the prime number
theorem,

$\sum_{p\leq Y}\frac{f_{\theta}(p)}{p}=\sum_{p\leq
Y}\frac{2}{p}+O\left(\theta^{2}\sum_{p\leq e^{1/\theta}}\frac{(\log
p)^{2}}{p}\right)=2\log(\min\\{1/\theta,\log X\\})+O(1).$

This proves the first claim if $0\leq\theta\leq 1/\log X$, so assume now that
$1/\log X<\theta\leq 1$. By partial summation and the prime number theorem, we
have

$\displaystyle\sum_{Y<p\leq X}\frac{1+\cos(\theta\log
p)}{p}=\int_{1}^{\theta\log X}(1+\cos v)\frac{dv}{v}+O(1)$
$\displaystyle=\left(\frac{1}{2\pi}\int_{0}^{2\pi}(1+\cos
u)du\right)\log(\theta\log X)+O(1)$ $\displaystyle=\log(\theta\log X)+O(1).$

We thus deduce that

$\sum_{p\leq X}\frac{f_{\theta}(p)}{p}=2\log(1/\theta)+\log(\theta\log
X)=\log(\theta^{-1}\log X),$

and the first claim follows for all $\frac{1}{\log X}<\theta\leq 1$ as well.
For the second claim, we simply note that

$H(f_{\theta};X)\asymp\exp\left(\sum_{p\leq
X}\frac{(f_{\theta}(p)-1)^{2}}{p}\right)=\exp\left(\sum_{p\leq
X}\frac{\cos(\theta\log p)^{2}}{p}\right).$

A similar partial summation argument shows that

$\displaystyle\sum_{p\leq X}\frac{\cos(\theta\log p)^{2}}{p}$
$\displaystyle=\log(\min\\{1/\theta,\log
X\\})+\left(\frac{1}{2\pi}\int_{0}^{2\pi}(\cos
u)^{2}du\right)\log(1+\theta\log X)+O(1)$
$\displaystyle=\frac{1}{2}\log(\min\\{\log X,1/\theta\\}\log X)+O(1),$

and the claim follows. ∎

######  Lemma 5.10.

Let $\theta\in(0,1]$. Let $2\leq z\leq w\leq X$. Then

$\sum_{z<p\leq w}\frac{f_{\theta}(p)}{p}\geq\sum_{z<p\leq
w}\frac{1}{p}+O(1/\log z).$

###### Proof.

Set $Y:=\min\\{X,e^{1/\theta}\\}$ once again. If $w\leq Y$ then
$\cos(\theta\log p)\geq\cos(1)\geq 0$ for all $z<p\leq w$, and thus

$\sum_{z<p\leq w}\frac{f_{\theta}(p)}{p}\geq\sum_{z<p\leq
w}\frac{1}{p}+O(1/\log z).$

On the other hand, if $Y\leq z$ (so that $\theta>1/\log X$) then by the same
partial summation argument as in Lemma 5.9 we find that

$\sum_{z<p\leq
w}\frac{f_{\theta}(p)}{p}=\left(\frac{1}{2\pi}\int_{0}^{2\pi}(1+\cos
u)du\right)\log(\log w/\log z)+O(1/\log z)=\sum_{z<p\leq
w}\frac{1}{p}+O(1/\log z).$

Finally, suppose $z<Y<w$. In this case, we split the interval into the
segments $(z,Y]$ and $(Y,w]$ and apply the arguments in each of the previous
two cases to obtain

$\sum_{z<p\leq w}\frac{f_{\theta}(p)}{p}\geq\sum_{z<p\leq
Y}\frac{1}{p}+\sum_{Y<p\leq w}\frac{1}{p}+O(1/\log z)\geq\sum_{z<p\leq
w}\frac{1}{p}+O(1/\log z),$

as claimed. ∎

###### Proof of Corollary 5.8.

Let $h=h_{0}(\theta^{-1}\log X)$. Note that $|d(n;\theta)|^{2}/d(n)\leq d(n)$
uniformly over $n$, so that combined with Lemma 5.10 we have that
$f_{\theta}\in\mathcal{M}(X;1,2,1)$ for all $\theta\in(1/\log X,1]$. As in
Lemma 5.4, $t_{0}(f_{\theta},X)=0$ is admissible for all $\theta\in(1/\log
X,1]$. Since $H(f_{\theta};X)\ll(\theta^{-1}\log X)^{1/2}$ uniformly over all
$\theta\in(1/\log X,1]$ by Lemma 5.9, the claim follows from Corollary 1.8. ∎

###### Proof of Corollary 1.6.

Let $\delta\in(0,1]$, set $Y:=\exp\left((\log X)^{\delta}\right)$ and put
$h=h_{0}(\log X)^{(1+\delta)/2}$, with $10\leq h_{0}\leq X/10(\log
X)^{(1+\delta)/2}$. By Lemma 5.9 we have $H(f_{\theta};X)\gg(\log
X)^{(1+\delta)/2}$ for all $1/\log Y<\theta\leq 1$. By Fubini’s theorem, the
Cauchy-Schwarz inequality and Corollary 5.8, we thus see that

$\displaystyle\frac{2}{X}\int_{2/X}^{X}\left[\int_{1/\log
Y}^{1}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)-\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right|d\theta\right]dx$ $\displaystyle\leq\int_{1/\log
Y}^{1}\left(\frac{2}{X}\int_{X/2}^{X}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)-\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right|^{2}dx\right)^{1/2}d\theta$
$\displaystyle\ll\left(\sqrt{\frac{\log\log h_{0}}{\log
h_{0}}}+\left(\frac{\log\log X}{(\log
X)^{\kappa}}\right)^{\frac{1}{2}}\right)\int_{1/\log
Y}^{1}\mathcal{P}_{f_{\theta}}(X)d\theta,$

for any $0<\kappa<\kappa_{1,2}$. By Lemma 5.9 we have

$\int_{1/\log Y}^{1}\mathcal{P}_{f_{\theta}}(X)d\theta\ll\int_{1/\log
Y}^{1}\exp\left(\sum_{p\leq
X}\frac{f_{\theta}(p)-1}{p}\right)d\theta\ll\int_{1/\log
Y}^{1}\frac{d\theta}{\theta}=\log\log Y.$

We thus deduce that for all but $o_{h_{0}\rightarrow\infty}(X)$ exceptional
integers $x\in[X/2,X]$, we have that

$\int_{1/\log Y}^{1}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)-\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right|d\theta=o_{h_{0}\rightarrow\infty}(\log\log Y).$

For any of the non-exceptional $x$, we apply (29) to give

$\displaystyle\frac{1}{h}\sum_{x-h<n\leq
x}\Delta(n)\geq\int_{0}^{1}\left(\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)\right)d\theta$ $\displaystyle\geq\int_{1/\log
Y}^{1}\left(\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right)d\theta-\int_{1/\log
Y}^{1}\left|\frac{1}{h}\sum_{x-h<n\leq
x}f_{\theta}(n)-\frac{2}{X}\sum_{X/2<n\leq X}f_{\theta}(n)\right|d\theta$ (30)
$\displaystyle=\int_{1/\log Y}^{1}\left(\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right)d\theta-o_{h_{0}\rightarrow\infty}(\log\log Y).$

On the other hand, by [41, Exer. 208] we find that when $1/\log
X\leq\theta\leq 1$,

$\frac{2}{X}\sum_{X/2<n\leq
X}\frac{|d(n,\theta)|^{2}}{d(n)}\geq\frac{2}{X}\sum_{X/2<n\leq
X}\frac{\mu^{2}(n)|d(n,\theta)|^{2}}{d(n)}=|\zeta(1+i\theta)|H_{\theta}(1)+O(|\theta^{3/2}|/\sqrt{\log
X}),$

where for $\text{Re}(s)>3/4$, $H_{\theta}(s)$ is some convergent Dirichlet
series satisfying $H_{\theta}(1)\gg 1$ uniformly in $\theta\in[1/\log X,1]$.
Integrating over $\theta\in[1/\log Y,1]$ and using the Laurent expansion
$\zeta(1+i\theta)=(i\theta)^{-1}+O(1)$, we deduce that

(31) $\int_{1/\log Y}^{1}\left(\frac{2}{X}\sum_{X/2<n\leq
X}f_{\theta}(n)\right)d\theta\gg\int_{1/\log
Y}^{1}\frac{d\theta}{\theta}+O(1)=\delta\log\log X+O(1).$

We thus have obtained

$\frac{1}{h}\sum_{x-h<n\leq x}\Delta(n)\gg\delta\log\log X$

for all but $o_{h_{0}\rightarrow\infty}(X)$ integers $x\in[X/2,X]$, and the
claim follows. ∎

## Acknowledgments

The author warmly thanks Oleksiy Klurman and Aled Walker for helpful
suggestions about improving the exposition of the paper, as well as for their
encouragement. He is also grateful to Maksym Radziwiłł and Jesse Thorner for
helpful conversations and suggestions regarding the applications to
automorphic forms. Finally, he would like to thank Gérald Tenenbaum for useful
comments and references. Much of this paper was written while the author held
a Junior Fellowship at the Mittag-Leffler institute for mathematical research
during the Winter of 2021. He would like to thank the institute for its
support.

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