# On Equal Consecutive Values of Multiplicative Functions

Alexander P. Mangerel  Department of Mathematical Sciences, Durham University,
Stockton Road, Durham, DH1 3LE, UK<EMAIL_ADDRESS>
###### Abstract.

Let $f:\mathbb{N}\rightarrow\mathbb{C}$ be a multiplicative function for which

$\sum_{p:\,|f(p)|\neq 1}\frac{1}{p}=\infty.$

We show under this condition alone that for any integer $h\neq 0$ the set

$\\{n\in\mathbb{N}:f(n)=f(n+h)\neq 0\\}$

has logarithmic density 0. We also prove a converse result, along with an
application to the Fourier coefficients of holomorphic cusp forms.
The proof involves analysing the value distribution of $f$ using the
compositions $|f|^{it}$, relying crucially on various applications of Tao’s
theorem on logarithmically-averaged correlations of non-pretentious
multiplicative functions. Further key inputs arise from the inverse theory of
sumsets in continuous additive combinatorics.

##  1\. Introduction

There is an expected general phenomenon in analytic number theory that the
factorisations of additively coupled positive integers $n$ and $n+h$, $h\geq
1$, are _statistically independent_ , in the sense that for _very few_
$n\in\mathbb{N}$ do we expect the features (number of primes, size of prime
factors etc.) of the prime factorisation of $n$ to influence the corresponding
features of the factorisation of $n+h$. To investigate this phenomenon we are
motivated to study the joint value distribution, as $n$ varies, of pairs
$(f(n),f(n+h))$, where $f:\mathbb{N}\rightarrow\mathbb{C}$ is a multiplicative
function.

Bounding the size of the set $\\{n\leq x:f(n)=f(n+h)\\}$ as
$x\rightarrow\infty$, for $h\in\mathbb{N}$ and $f$ a multiplicative function,
is a natural problem, for which some precedent in the literature already
exists. For example, when $f=d$, the divisor function, Erdős, Pomerance and
Sarkőzy [4] proved using sieve theoretic arguments that

$|\\{n\leq x:d(n)=d(n+1)\\}|\ll\frac{x}{\sqrt{\log\log x}},$

and thus $d(n)=d(n+1)$ occurs on a set of natural111The _density_ of a set
$A\subseteq\mathbb{N}$ is defined by
$\lim_{x\rightarrow\infty}|A\cap[1,x]|/x$, provided the limit exists. The
_upper density_ of $f$ is given by the same definition with $\limsup$ in place
of $\lim$. density $0$. In a separate paper [3], the same authors obtain
stronger upper bounds for the number of solutions $n\leq x$ to
$\sigma(n)=\sigma(n+1)$ and to $\phi(n)=\phi(n+1)$, where $\sigma$ and $\phi$
are, respectively, the sum-of-divisors function and Euler’s phi function. The
explicit nature of these three examples is significant in the arguments of [4]
and [3].

In a recent paper [8], Klurman and the author considered a different example
of interest, when $(f(n))_{n}$ corresponds to the sequence of Fourier
coefficients of a holomorphic cusp form. More precisely, for $k\geq 2$ let
$\phi$ be a weight $k$, arithmetically normalised, holomorphic cuspidal
eigenform without complex multiplication for $\text{SL}_{2}(\mathbb{Z})$.
Writing $\phi$ in its Fourier expansion

$\phi(z)=\sum_{n\geq 1}a_{\phi}(n)e(nz),\quad\text{Im}(z)>0,$

normalised so that $a_{\phi}(1)=1$, it was shown among other things that each
of the sets

$\\{n\in\mathbb{N}:|a_{\phi}(n)|<|a_{\phi}(n+h)|\\},\quad\\{m\in\mathbb{N}:|a_{\phi}(m+h)|<|a_{\phi}(m)|\\},$

has positive density, and in fact it follows from [8, Thm. 1.7] that for every
$h\geq 1$ the set of _non-vanishing_ coincidences222The non-vanishing
assumption here is necessary given our current knowledge about the vanishing
of Fourier coefficients of cusp forms. In particular, if there exists even a
single _odd_ integer $d_{0}$ for which $a_{\phi}(d_{0})=0$ then a positive
density sequence of coincidences $a_{\phi}(n+d_{0})=a_{\phi}(n)=0$ can be
found. Indeed, it suffices to choose the set $\\{md_{0}:m\in\mathbb{N},\
(m(m+1),d_{0})=1\\}$. Ruling out the existence of such $d_{0}$ for the
Ramanujan $\tau$-function, for example, would require proving a well-known
conjecture due to Lehmer.

$\\{n\in\mathbb{N}:\,a_{\phi}(n)=a_{\phi}(n+h)\neq 0\\}$

has natural density $0$. This can be understood heuristically from the fact
that if $\phi$ has weight $k\geq 2$ then $a_{\phi}(n)$ generically grows like
$n^{(k-1)/2+o(1)}$ (due to work of Deligne), and thus the likelihood of
coincidences among non-zero values in such a large set is rather small.

More generally, we might speculate (as is done in some form on [8, p. 2]) that
the same conclusion about the paucity of coincidences $f(n)=f(n+h)$ ought to
be true for _any_ unbounded multiplicative function
$f:\mathbb{N}\rightarrow\mathbb{C}$ such that $|f(p)|\neq 1$ for most primes
$p$ (if, conversely, $|f(p)|=1$ for many $p$ this would account for numerous
coincidences among non-zero values in $(f(m))_{m}$, as indicated by
Proposition 1.5 below).

In this paper, we consider the general problem of bounding the number of
solutions in positive integers $n$ to the equation

$f(n)=f(n+h)\neq 0,$

where $f:\mathbb{N}\rightarrow\mathbb{C}$ is an _arbitary_ multiplicative
function, provided only that $|f(p)|\neq 1$ sufficiently often (in a precise
sense). While we are unable to establish that this set has _natural_ density
$0$, we do prove that it is a thin set in a weaker sense.

Our main result is the following.

######  Theorem 1.1.

Let $f_{1},f_{2}:\mathbb{N}\rightarrow\mathbb{C}$ be multiplicative functions
and let $h\geq 1$. Assume that

(1) $\displaystyle\sum_{p:|f_{1}(p)|\neq 1}\frac{1}{p}=\infty.$

Let $a,b\in\mathbb{N}$, $c,d\in\mathbb{Z}$ with $ad-bc\neq 0$ and
$(a,b)=(c,d)=1$. Let also $C\in\mathbb{C}\backslash\\{0\\}$. Then the set

$\\{n\in\mathbb{N}:f_{1}(an+b)=Cf_{2}(cn+d)\neq 0\\}$

has logarithmic density $0$, i.e., we have

(2) $\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq x\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}=o(1).$

######  Remark 1.2.

Ideally, we would like to prove the same result with the notion of
“logarithmic density” replaced by “natural density”. Unfortunately, we are
(unconditionally) limited in this matter by the fact that the linchpin of our
proof, Tao’s theorem on logarithmically-averaged binary correlations of
multiplicative functions (see Theorem 2.2 below) has not been proven for
Cesàro-averaged binary correlations. The corresponding result is, however,
expected to hold by a conjecture due to Elliott (see [17] for a discussion),
and assuming Elliott’s conjecture our result would immediately be upgraded in
this way.

While a Cesàro-averaged result of this kind is known at _almost every scale_
$x$ (see [18, Cor. 1.13]), this variant would be incompatible with our
objectives, since in order to prove Theorem 1.1 we already need to restrict
our attention to an arbitrary infinite subsequence of scales which may not
intersect with the “good” scales of such a theorem.

######  Remark 1.3.

The non-vanishing condition $f_{1}(an+b)f_{2}(cn+d)\neq 0$ is necessary333We
thank the anonymous referee for pointing out this simple construction. in this
theorem. Indeed, take any complex-valued multiplicative function $f$ and
define $\tilde{f}:=\mu^{2}f$. Then $\tilde{f}$ is supported on squarefree
integers. For any fixed, distinct primes $p_{1},p_{2}$ not dividing $ac(ad-
bc)$ the set

$\\{n\in\mathbb{N}:\tilde{f}(an+b)=\tilde{f}(cn+d)\\}$

contains the positive density set

$\\{n\in\mathbb{N}:p_{1}^{2}|(an+b),\quad p_{2}^{2}|(cn+d)\\}.$

For any $n$ in this subset both $an+b$ and $cn+d$ are _not_ squarefree, and
hence $\tilde{f}(an+b)=0=\tilde{f}(cn+d)$.

######  Remark 1.4.

When $f_{1},f_{2}$ are completely multiplicative, the condition
$(a,b)=(c,d)=1$ may be dropped, as we may always reduce to the coprime case
(by adjusting the value of $C$).

Theorem 1.1 can be compared with the main result of [2]. There, Elliott and
Kish prove that if $f:\mathbb{N}\rightarrow\mathbb{C}$ is completely
multiplicative and

$f(an+b)=Cf(cn+d)\neq 0\text{ for \emph{all} }n\geq n_{0},$

where $ad-bc\neq 0$ and $C\in\mathbb{C}\backslash\\{0\\}$, then there is a
Dirichlet character $\chi$ of some modulus $m=m(a,b,c,d)$ such that

$f(p)=\chi(p)\text{ for all }p\nmid m.$

When $f_{1}=f_{2}=f$ our result addresses the “$1\%$ world” version of their
problem, showing that if the (logarithmic) proportion of $n$ that satisfy the
equation is $\geq\delta$, for $\delta>0$ a small parameter, then

(3) $\sum_{p:|f(p)|\neq 1}\frac{1}{p}<\infty.$

While this is consistent with the result in [2], in our setting no further
structural information about $f$ can possibly be gleaned, and in particular
$f$ need not behave like a Dirichlet character, or even be pretentious (see
Section 2.1 for the relevant definition)! Indeed, it is known for instance
[17, Cor. 1.7] that the Möbius function $\mu$ satisfies

$\mu(n)=\mu(n+1)\neq 0\text{ on a set of logarithmic density
}\frac{1}{2}\prod_{p}(1-2/p^{2}),$

and it is well-known that $\mu$ is non-pretentious. In fact, we may obtain the
following (elementary) converse for real-valued $f$; for simplicity we
constrain ourselves to a single case of affine maps $an+b$ and $cn+d$.

######  Proposition 1.5.

Let $f:\mathbb{N}\rightarrow\mathbb{R}\backslash\\{0\\}$ be a multiplicative
function for which (3) holds. Then there is a $u\in\\{-1,+1\\}$ such that

$\limsup_{x\rightarrow\infty}\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq
x\end{subarray}}\frac{1_{f(n)=uf(n+2)}}{n}>0.$

Theorem 1.1 allows us to recover the result

(4) $\lim_{x\rightarrow\infty}\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq
x\\\ a_{\phi}(n)a_{\phi}(n+h)\neq
0\end{subarray}}\frac{1_{a_{\phi}(n)=a_{\phi}(n+h)}}{n}=0$

for the sequence of Fourier coefficients $(a_{\phi}(n))_{n}$ of any
holomorphic cusp form without CM for $\text{SL}_{2}(\mathbb{Z})$. While this
is weaker than what is proved in [8], (4) may be shown _without_ appealing to
an effective version of the Sato-Tate theorem such as that found in [19]. As
such, we may avoid having to apply the deep theorems of Newton-Thorne [14] on
the functoriality of symmetric power lifts of holomorphic cusp forms. Indeed,
our proof (see Section 5) requires nothing more than:

  * •

the prime number theorem for Rankin-Selberg $L$-functions, and

  * •

Deligne’s bound for the prime-indexed Fourier coefficients $|a_{\phi}(p)|$,

both of which apply (at least conjecturally and in some cases unconditionally)
to the sequence of coefficients of the standard $L$-function of a broader
collection of cuspidal automorphic forms.

######  Corollary 1.6.

Let $h\geq 1$ be fixed and let $\phi_{1},\phi_{2}$ be holomorphic cusp forms
for the full modular group. Then the set

$\\{n\in\mathbb{N}:a_{\phi_{1}}(n)=a_{\phi_{2}}(n+j)\neq 0\text{ for some
}1\leq|j|\leq h\\}$

has logarithmic density 0.

######  Remark 1.7.

Note that the claim is trivial if, say, $\phi_{1}$ is assumed to have CM. In
this case, it is known [16] that

$|\\{n\leq X:a_{\phi_{1}}(n)\neq 0\\}|\asymp\frac{X}{\sqrt{\log X}},$

which necessarily implies that
$\\{n\in\mathbb{N}:a_{\phi_{1}}(n)=a_{\phi_{2}}(n+h)\neq 0\\}$ has logarithmic
density $0$.

######  Remark 1.8.

A result like Corollary 1.6 may be proved for the sequences of coefficients
$(a_{\pi_{j}}(n))_{n}$ of the standard $L$-functions of fixed cuspidal
automorphic representations $\pi_{j}$ for $\text{GL}_{m_{j}}$, $m_{j}\geq 1$,
with unitary central character ($j=1,2$), assuming that at least one of
$\pi_{1},\pi_{2}$ satisfies the Generalised Ramanujan Conjecture (see e.g.
[13, Sec. 1.3.1] for relevant preliminaries). We leave this extension to the
interested reader.

###  1.1. Proof Ideas

####  1.1.1. Proof strategy towards Proposition 1.5

We prove Proposition 1.5 in Section 4 by studying the equation
$|f(n)|=|f(n+2)|$ for _squarefree_ $n$ and $n+2$, noting that since $f$ is
real-valued this implies that one of $f(n)=f(n+2)$ or $f(n)=-f(n+2)$ holds.
Now, as $f$ is multiplicative and $n$ and $n+2$ are squarefree,

$|f(n)|=|f(n+2)|\text{ whenever }|f(p)|=1\text{ for all }p|n(n+2).$

Writing $\mathcal{P}_{\neq 1}:=\\{p:|f(p)|\neq 1\\}$, it thus suffices to show
that the set444For a set $S\subset\mathbb{N}$ we write $(n,S)=1$ if $(n,m)=1$
for all $m\in S$.

$\\{n\in\mathbb{N}:(n(n+2),\mathcal{P}_{\neq 1})=1\\}$

has positive density. This follows straightforwardly from a zero-dimensional
sieve, using crucially the fact that $\sum_{p\in\mathcal{P}_{\neq
1}}p^{-1}<\infty$.

####  1.1.2. Proof strategy towards Theorem 1.1

The proof of Theorem 1.1 is the main goal of the paper. We will assume for the
sake of contradiction the existence of a (small) parameter $\delta\in(0,1)$
and an infinite increasing sequence $(x_{j})_{j}$ such that

(5) $\frac{1}{\log x_{j}}\sum_{\begin{subarray}{c}n\leq x_{j}\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}\geq\delta,\quad j\geq
1.$

We seek a contradiction by obtaining from (5) precise information about the
sequence $(|f(p)|)_{p}$.

In order to analyse the condition (5) we appeal to a deep theorem of Tao [17]
(see Theorem 2.2 below) on logarithmically-averaged correlations of
multiplicative functions $g_{1},g_{2}$ taking values in $\mathbb{U}$, the
closed unit disc. This result implies structural information about the prime-
indexed sequences $(g_{j}(p))_{p}$, $j=1,2$, whenever

$\lim_{x\rightarrow\infty}\frac{1}{\log x}\left|\sum_{n\leq
x}\frac{g_{1}(an+b)g_{2}(cn+d)}{n}\right|\neq 0.$

To apply this we “project” our multiplicative functions $f_{1},f_{2}$ onto
$\mathbb{U}$ in a manner that retains their multiplicativity.

For ease of exposition, suppose that our affine linear forms are
$(an+b,cn+d)=(n,n+h)$, $h\geq 1$, that $C=1$ and that $f_{1}=f_{2}=f$. We
observe that for $t\in\mathbb{R}$ the composition

$|f|_{t}(n):=\begin{cases}|f(n)|^{it}&\text{ if $f(n)\neq 0$}\\\ 0&\text{ if
$f(n)=0$}\end{cases}$

is a well-defined multiplicative function that takes values in $\mathbb{U}$.
Let $A\geq 1$ be a parameter to be chosen later. The function

$g(n):=A\log|f(n)|$

is well-defined on the set $\\{n\in\mathbb{N}:f(n)\neq 0\\}$, and is additive
when restricted to that set. In particular, we have

$f(n)=f(n+h)\neq 0\Rightarrow g(n)=g(n+h),$

and also555Here we use the standard notation $e(y):=e^{2\pi iy}$ for
$y\in\mathbb{R}$. $|f|_{2\pi At}(n)=e(tg(n))$. Using a Fourier analytic
identity, we thus show that if (5) holds then there is a bounded set
$X\subset\mathbb{R}$ with positive Lebesgue measure such that for each $y\in
X$ we have lower bounds

$\frac{1}{\log x_{k}}\left|\sum_{n\leq x_{k}}\frac{|f|_{2\pi Ay}(n)|f|_{-2\pi
Ay}(n+h)}{n}\right|\gg_{\delta}1.$

For each of these $y\in X$, Tao’s theorem implies the existence of a Dirichlet
character $\chi_{y}$ of conductor depending only on $\delta$, and
$t_{y}\in\mathbb{R}$ of controlled size such that $f_{y}$ pretends to be a
twisted character $\chi_{y}(n)n^{it_{y}}$ (see Section 2.1 for a relevant
definition). By passing to a positive measure subset $X^{\prime}$ of $X$ if
needed we may assume that $\chi_{y}$ is the same for all $y$, and so we focus
on studying the dependence of $t_{y}$ on $y$.

Using an idea going back to Halász [6] and subsequently refined by Ruzsa [15],
we show that the mapping $y\mapsto t_{y}$ extends to an entire interval $I$
and is uniformly approximated on $I$ by a linear function $y\mapsto ry$ for
some $r\in\mathbb{R}$ depending at most on $\delta$. In this way we find that
for most primes $p\leq x_{k}$ and for all $y\in I$,

$|f(p)|^{2\pi iAy}\approx p^{2\pi it_{y}}\approx p^{2\pi iry},$

and thus that $|f(p)|^{A}\approx p^{r}$ for “typical” primes $p$. Since $A$ is
arbitrarily large, we deduce that $|f(p)|=1$ outside of a sparse set of
primes, which contradicts our assumption (1).

######  Remark 1.9.

In the recent work [11], we considered the variant problem of bounding the
size of sets

$\\{n\leq x:f(n+a)=f(n)+b\\},\text{ where }ab\neq 0,\,x\geq 1,$

for $f$ an _integer-valued_ multiplicative function. We used this together
with the work in the present paper to obtain a partial classification of all
unbounded multiplicative functions $f:\mathbb{N}\rightarrow\mathbb{N}$ for
which the set

$\\{n\in\mathbb{N}:|f(n+1)-f(n)|\leq C\\},$

i.e., the set of $n$ for which the gaps between consecutive values of $f(n)$
is bounded by some $C>0$, has positive upper density.

Unlike in the _homogeneous_ (i.e., $a=0$) problem considered in this paper,
examples do exist where the number of such solutions has positive density
(e.g., take $a=b=1$, then $f(n)=n$ is such an example). In addition to the
“archimedean” projections $|f(n)|^{it}$ applied here, we also consider
corresponding “non-archimedean” projections $\chi(f(n))$, for $\chi$ a
Dirichlet character to a suitably chosen modulus $q\geq 1$. The application of
these latter projections were developed in connection to the present paper,
but thanks to a neat observation by the anonymous referee these turned out to
be unnecessary here.

### Acknowledgments

The author would like to warmly thank the anonymous referee for their valuable
suggestions on a previous version of this paper, leading to a shorter proof of
Theorem 1.1 with wider applicability.

##  2\. Background Results about Multiplicative Functions

In this section we collect some background notions and results about
multiplicative functions and pretentious number theory to be used in the
sequel.

###  2.1. Properties of the pretentious distance

A key rôle is played by the pretentious distances (in the sense of Granville
and Soundararajan). Precisely, for arithmetic functions
$f,g:\mathbb{N}\rightarrow\mathbb{U}$ and $x\geq 2$, define

$\mathbb{D}(f,g;x):=\left(\sum_{p\leq
x}\frac{1-\text{Re}(f(p)\overline{g}(p))}{p}\right)^{1/2},\quad\mathbb{D}(f,g;\infty):=\lim_{x\rightarrow\infty}\mathbb{D}(f,g;x).$

Clearly, $0\leq\mathbb{D}(f,g;x)^{2}\leq 2\log\log x+O(1)$ by Mertens’
theorem. When $\mathbb{D}(f,g;\infty)<\infty$ (so that the upper bound
$2\log\log x+O(1)$ is far from the truth) we say that $f$ _pretends to be_ $g$
(symmetrically, $g$ _pretends to be_ $f$), or that $f$ is _$g$ -pretentious_.
The pretentious distance functions satisfy the _pretentious triangle
inequality_ , which we will use in the following two forms [5, Lem. 3.1]. For
functions $f,g,h,f_{1},f_{2},g_{1},g_{2}:\mathbb{N}\rightarrow\mathbb{U}$, we
have

(6) $\displaystyle\mathbb{D}(f,h;x)\leq\mathbb{D}(f,g;x)+\mathbb{D}(g,h;x),$
(7)
$\displaystyle\mathbb{D}(f_{1}f_{2},g_{1}g_{2};x)\leq\mathbb{D}(f_{1},g_{1};x)+\mathbb{D}(f_{2},g_{2};x).$

Applying (7) inductively, we can show that for
$f,g:\mathbb{N}\rightarrow\mathbb{U}$ and $m\geq 1$,

(8) $\mathbb{D}(f^{m},g^{m};x)\leq m\mathbb{D}(f,g;x).$

This result implies, for instance, that if $f$ pretends to be $g$ then $f^{m}$
pretends to be $g^{m}$. The latter is especially helpful when, say, $g$ takes
its non-zero values in roots of unity of some order (as is the case for
instance for Dirichlet characters), in which case $g^{m}=|g|$, and thus
$f^{m}$ pretends to be a function taking values $0$ or $1$ (and thus is, in a
precise sense, “close” to being non-negative).
Note that when $f(p),g(p)\in S^{1}$ we may rewrite the numerators of the
summands in the definition of $\mathbb{D}(f,g;x)$ using the relation

$|f(p)-g(p)|^{2}=2(1-\text{Re}(f(p)\overline{g}(p))).$

In particular, it follows that if $f$ is $g$-pretentious and $\eta>0$ then the
set

$\mathcal{G}_{\eta}:=\\{p:|f(p)-g(p)|>\eta\\}$

satisfies

$\sum_{\begin{subarray}{c}p\leq x\\\
p\in\mathcal{G}_{\eta}\end{subarray}}\frac{1}{p}\leq\eta^{-2}\sum_{p\leq
x}\frac{|f(p)-g(p)|^{2}}{p}=\eta^{-2}\mathbb{D}(f,g;x)^{2}\ll_{\eta}1.$

For convenience, in the sequel we will refer to a subset of primes $S$ as
being _sparse_ (respectively _$C$ -sparse_) if

$\sum_{\begin{subarray}{c}p\leq x\\\ p\in S\end{subarray}}\frac{1}{p}\ll 1,$

(respectively, $\ll_{C}1$) as $x\rightarrow\infty$.

The Archimedean characters $n\mapsto n^{it}$, $t\in\mathbb{R}$, form an
important class of examples of multiplicative functions with non-zero mean
values and correlations (see Theorem 2.2 below, for example). In many results
in the sequel we will need control of the distance between $n^{it}$ and $1$.
This is provided by the following standard result.

######  Lemma 2.1.

Let $\varepsilon>0$ and let $x\geq x_{0}(\varepsilon)$. If $10\leq|t|\leq
x^{2}$ then

(9) $\mathbb{D}(n^{it},1;x)^{2}\geq(1/3-\varepsilon)\log\log x.$

Moreover, if $|t|\leq 10$ then

(10) $\mathbb{D}(n^{it},1;x)^{2}=\log(1+|t|\log x)+O(1).$

###### Proof.

Since we could not locate a proof of this precise statement in the literature,
we give one here.
Set $\sigma:=1+1/\log x$. Note that for any $t\in\mathbb{R}$, Mertens’ theorem
implies that

$\mathbb{D}(1,n^{it};x)^{2}=\log\log x-\sum_{p\leq
x}\frac{\text{Re}(p^{it})}{p}+O(1)=\log\log x-\log|\zeta(\sigma+it)|+O(1),$

When $|t|\leq 10$ the Laurent expansion of $\zeta$ near $s=1$ yields

$\zeta(\sigma+it)=\frac{1}{\sigma-1+it}+O(1+|\sigma+it-1|)=\frac{\log
x}{1+it\log x}+O(1),$

so that since $(1+|t|\log x)^{2}\leq 2(1+t^{2}\log^{2}x)\leq 2(1+|t|\log
x)^{2}$ we find

$\log|\zeta(\sigma+it)|=\log\log
x-\frac{1}{2}\log(1+t^{2}\log^{2}x)+O(1)=\log\log x-\log(1+|t|\log x)+O(1).$

This proves (10) in the case $|t|\leq 10$.
Next, suppose that $10\leq|t|\leq x^{2}$. Set $y:=V_{x^{2}}\geq V_{t}$, where

$V_{t}:=\exp((\log^{2/3}(2+|t|)\log\log^{1/3}(2+|t|)),\quad t\in\mathbb{R}.$

For $\text{Re}(s)>1$ define

$\zeta_{y}(s):=\prod_{p>y}\left(1-\frac{1}{p^{s}}\right)^{-1}.$

Combining [9, Lem. 3.2] and [9, Lem. 4.1] (with $\chi$ the trivial character),
we find that

$\text{Re}\left(\sum_{y<p\leq
x}\frac{1}{p^{1+it}}\right)=\log|\zeta_{y}(\sigma+it)|+O(1)=O(1).$

Therefore, we find

$\mathbb{D}(1,n^{it};x)^{2}\geq\sum_{y<p\leq
x}\frac{1-\text{Re}(p^{-it})}{p}=\log\left(\frac{\log x}{\log
y}\right)+O(1)=\frac{1}{3}\left(\log\log x-\log\log\log x\right)+O(1),$

and the claim follows. ∎

###  2.2. Tao’s theorem on logarithmically-averaged correlations

As described in the previous section, a key tool in our analysis of the
distribution of pairs $(f(an+b),f(cn+d))$ is the following theorem of Tao
[17], which prescribes necessary conditions for two multiplicative function
$g_{j}:\mathbb{N}\rightarrow\mathbb{U}$, $j=1,2$, to have large,
logarithmically-averaged binary correlations.

######  Theorem 2.2 (Tao’s theorem on binary correlations).

Let $\eta>0$, and let $a,c\in\mathbb{N}$, $b,d\in\mathbb{Z}$ with $ad-bc\neq
0$. Let $g_{1},g_{2}:\mathbb{N}\rightarrow\mathbb{U}$ be 1-bounded
multiplicative functions with the property that for some $x\geq x_{0}(\eta)$
we have

$\frac{1}{\log x}\left|\sum_{n\leq
x}\frac{g_{1}(an+b)g_{2}(cn+d)}{n}\right|\geq\eta.$

Then for each $j=1,2$ there is a real number $t_{j}=t_{j,x}$ with
$t_{j}=O_{\eta}(x)$ and a primitive Dirichlet character $\psi_{j}=\psi_{j,x}$
with conductor $O_{\eta}(1)$ such that

$\sum_{p\leq
x}\frac{1-\mathrm{Re}(g_{j}(p)\overline{\psi}_{j}(p)p^{-it_{j}})}{p}=O_{\eta}(1).$

##  3\. The archimedean projections and the proof of Theorem 1.1

Using Tao’s theorem as a principal tool, we will prove the following.

######  Proposition 3.1.

Let $f_{1},f_{2}:\mathbb{N}\rightarrow\mathbb{C}$ be multiplicative functions,
and suppose $f_{1}$ satisfies

(11) $\sum_{p:f_{1}(p)=0}\frac{1}{p}<\infty,\quad\sum_{p:|f_{1}(p)|\neq
1}\frac{1}{p}=\infty.$

Let $a,c\in\mathbb{N}$, $b,d\in\mathbb{Z}$ and $C\in\mathbb{C}$, with $C(ad-
bc)\neq 0$. Suppose there is a $\delta>0$ and an infinite increasing sequence
$(x_{j})_{j}$ such that for each $j\geq 1$,

(12) $\frac{1}{\log x_{j}}\sum_{\begin{subarray}{c}n\leq x_{j}\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}\geq\delta.$

Then if $j\geq j_{0}(\delta)$ and $A\geq 1$ there is a parameter
$r_{j}=r_{j}(A)\in[-x_{j}^{2},x_{j}^{2}]$ such that if $\delta^{-2}\leq
B\ll_{\delta}1$ then

(13) $\sum_{\begin{subarray}{c}p\leq x_{j}\\\
|f_{1}(p)|^{A}/p^{r_{j}}\notin[e^{-2/B},e^{2/B}]\end{subarray}}\frac{1}{p}\ll_{\delta}1,$

the bound (13) being independent of $A$.

######  Remark 3.2.

As is suggested by the hypotheses of Proposition 3.1, we will be able to
assume in the sequel that

(14) $\displaystyle\sum_{p:f_{1}(p)=0}\frac{1}{p}<\infty,$

as otherwise Theorem 1.1 is essentially trivial.
Indeed, as the function $1_{f_{1}(n)\neq 0}$ is multiplicative, it follows
that

$\sum_{n\leq x}\frac{1_{f_{1}(n)\neq 0}}{n}\ll\prod_{\begin{subarray}{c}p\leq
x\end{subarray}}\left(1+\frac{1_{f_{1}(p)\neq 0}}{p}\right)\ll(\log
x)\exp\left(\sum_{p\leq x}\frac{1_{f_{1}(p)\neq 0}-1}{p}\right)=o(\log x)$

whenever $\sum_{p:f_{1}(p)=0}p^{-1}=\infty$. This implies that
$\\{n:f_{1}(an+b)=Cf_{2}(cn+d)\neq 0\\}$ has logarithmic density $0$ for any
fixed $a,b,c,d,C$ as in the theorem.

Theorem 1.1 follows swiftly from Proposition 3.1.

###### Proof of Theorem 1.1.

Let $f_{1},f_{2}:\mathbb{N}\rightarrow\mathbb{C}$ be multiplicative functions
such that (1) holds. Assume for the sake of contradiction that

$\\{n\in\mathbb{N}:f_{1}(an+b)=Cf_{2}(cn+d)\neq 0\\}$

_does not_ have logarithmic density $0$, for some $a,c\in\mathbb{N}$,
$b,d\in\mathbb{Z}$ with $ad\neq bc$, $(a,b)=(c,d)=1$, and
$C\in\mathbb{C}\backslash\\{0\\}$. Then there is a $\delta>0$ and a sequence
of scales $(x_{j})_{j}$ such that for each $j\geq 1$,

$\frac{1}{\log x_{j}}\sum_{\begin{subarray}{c}n\leq x_{j}\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}\geq\delta,$

i.e., (12) holds. By Remark 3.2, it suffices to assume also that $f_{1}$
satisfies (14), and thus the conditions in (11) both hold by assumption.
Now, let $A\geq 1$. By Proposition 3.1, for each $j\geq j_{0}(\delta)$ there
exists $r_{j}=r_{j}(A)$ satisfying $|r_{j}|\leq x_{j}^{2}$ such that if we
define

$I_{A,j}:=[e^{-2/(AB)},e^{2/(AB)}x_{j}^{x_{j}^{2}/A}]$

then it follows that

$\Sigma_{A,j}:=\sum_{\begin{subarray}{c}p\leq x_{j}\\\ |f_{1}(p)|\notin
I_{A,j}\end{subarray}}\frac{1}{p}\leq\sum_{\begin{subarray}{c}p\leq x_{j}\\\
|f_{1}(p)|/p^{r_{j}/A}\notin[e^{-2/(AB)},e^{2/(AB)}]\end{subarray}}\frac{1}{p}\ll_{\delta}1,$

where the bound is independent of $A$. Since $I_{A^{\prime},j}\subseteq
I_{A,j}$ whenever $A^{\prime}\geq A$, it follows that $A\mapsto\Sigma_{A,j}$
is non-decreasing and uniformly bounded in $A$, for each $j$. Thus, the limit
$\Sigma_{j}:=\lim_{A\rightarrow\infty}\Sigma_{A,j}$ exists for each $j\geq
j_{0}(\delta)$, and since moreover

$\bigcap_{A\geq 1}I_{A,j}=\\{1\\}$

we deduce that

$\Sigma_{j}=\sum_{\begin{subarray}{c}p\leq x_{j}\\\ |f_{1}(p)|\neq
1\end{subarray}}\frac{1}{p}\ll_{\delta}1.$

Since $(\Sigma_{j})_{j}$ is also bounded and non-decreasing, it converges as
$j\rightarrow\infty$, and thus

$\sum_{p:|f_{1}(p)|\neq
1}\frac{1}{p}=\lim_{j\rightarrow\infty}\Sigma_{j}<\infty,$

which contradicts (1). The claim of the theorem thus follows. ∎

###  3.1. Reduction to correlations of $1$-bounded functions

We incorporate the maps $|f|_{t}:=|f|^{it}$ in proving the following, which
brings into play lower bounds for correlations of multiplicative functions
taking values in $\mathbb{U}$.

######  Lemma 3.3.

Let $A\geq 1$, $\delta\in(0,1)$ and let $B\geq\delta^{-2}$. Let $x$ be a
positive real number, sufficiently large with respect to $\delta$, for which

(15) $\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq x\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}\geq\delta.$

Define $F_{j}:=f_{j}^{A}$ for $j=1,2$. Then the set

(16) $X=X_{x,\delta,A,B}:=\Bigg{\\{}\alpha\in[-B,B]:\frac{1}{\log
x}\left|\sum_{\begin{subarray}{c}n\leq x\\\ f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{|F_{1}|_{\alpha}(an+b)|F_{2}|_{-\alpha}(cn+d)}{n}\right|\geq\frac{\delta}{16\pi}\Bigg{\\}}$

has Lebesgue measure $\geq\delta/(16\pi)$.

###### Proof.

Let $X$ denote the set in (16). Clearly, $X$ is symmetric and contains 0. The
equality $f_{1}(an+b)=Cf_{2}(cn+d)$ implies that
$|F_{1}(an+b)|=C^{\prime}|F_{2}(cn+d)|$, where $C^{\prime}:=|C|^{A}$. We next
relax this equality to an _inequality_. Borrowing an idea from Davenport and
Heilbronn [1], we observe that for $t\in\mathbb{R}$,

(17)
$\max\\{1-|t|,0\\}=\int_{-\infty}^{\infty}e(t\alpha)\left(\frac{\sin(\pi\alpha)}{\pi\alpha}\right)^{2}d\alpha.$

Crucially, this latter expression is absolutely integrable, with integral
$\leq 1$. Writing $g_{j}(n):=\log|F_{j}(n)|$, well-defined whenever
$f_{j}(n)\neq 0$, and $\rho:=\log|C^{\prime}|\in\mathbb{R}$, we have the
trivial implication

$\displaystyle f_{1}(an+b)=Cf_{2}(cn+d)\neq 0$
$\displaystyle\Rightarrow|g_{1}(an+b)-g_{2}(cn+d)-\rho|\leq 1/2,$

from which it follows immediately that

(18) $1_{f_{1}(an+b)=Cf_{2}(cn+d)\neq 0}\leq
2\max\\{1-|g_{1}(an+b)-g_{2}(cn+d)-\rho|,0\\}.$

Combining (18) with (17) and inserting the result into (15), we obtain

$\displaystyle\delta$ $\displaystyle\leq\frac{1}{\log
x}\sum_{\begin{subarray}{c}n\leq x\\\ f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{1_{f_{1}(an+b)=Cf_{2}(cn+d)}}{n}$ $\displaystyle\leq
2\int_{-\infty}^{\infty}\left(\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq
x\\\ f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{e(g_{1}(an+b)\alpha)e(-g_{2}(cn+d)\alpha)}{n}\right)e(-\rho\alpha)\left(\frac{\sin(\pi\alpha)}{\pi\alpha}\right)^{2}d\alpha$
$\displaystyle=2\int_{-\infty}^{\infty}\left(\frac{1}{\log
x}\sum_{\begin{subarray}{c}n\leq x\\\ f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{|F_{1}|_{2\pi\alpha}(an+b)|F_{2}|_{-2\pi\alpha}(cn+d)}{n}\right)e(-\rho\alpha)\left(\frac{\sin(\pi\alpha)}{\pi\alpha}\right)^{2}d\alpha.$

Bounding the range $|\alpha|>B$ trivially, we obtain after a change of
variables $\alpha\mapsto 2\pi\alpha$ that

(19) $\displaystyle\delta$ $\displaystyle\leq
4\pi\int_{-B}^{B}\left(\frac{1}{\log x}\sum_{\begin{subarray}{c}n\leq x\\\
f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{|F_{1}|_{\alpha}(an+b)|F_{2}|_{-\alpha}(cn+d)}{n}\right)e^{-i\rho\alpha}\left(\frac{\sin(\alpha/2)}{\alpha/2}\right)^{2}d\alpha+O\left(\frac{1}{B}\right).$

Recall that $B\geq\delta^{-2}$ and note that the integrand above is bounded by
$1$, From the definition (16) of $X$ and (19) we derive when $\delta$ is small
enough that

$\frac{\delta}{2}\leq 4\pi\int_{X}\left|\frac{1}{\log
x}\sum_{\begin{subarray}{c}n\leq x\\\ f_{1}(an+b)f_{2}(cn+d)\neq
0\end{subarray}}\frac{|F_{1}|_{\alpha}(an+b)|F_{2}|_{-\alpha}(cn+d)|}{n}\right|\left(\frac{\sin(\alpha/2)}{\alpha/2}\right)^{2}d\alpha+\frac{\delta}{4}\leq
4\pi\lambda(X)+\frac{\delta}{4},$

where $\lambda(X)$ denotes the Lebesgue measure of $X$. Rearranging, we obtain
$\lambda(X)\geq\delta/(16\pi)$, as claimed. ∎

As a corollary of this and Tao’s theorem, we deduce the following.

######  Corollary 3.4.

Assume the notation of Lemma 3.3, and let $f:\mathbb{N}\rightarrow\mathbb{C}$
be the multiplicative function given at prime powers by $f(p^{k})=1$ if
$f_{1}(p^{k})=0$, otherwise $f(p^{k})=F_{1}(p^{k})$. Then there are positive
integers $m,L=O_{\delta}(1)$, independent of $A$, such that for each $\beta\in
X$ there is a real number $t_{\beta}\in[-Lx,Lx]$ such that

$\mathbb{D}(|f^{m}|_{\beta},n^{imt_{\beta}};x)=O_{\delta}(1),$

and the upper bound is independent of $A$.

###### Proof.

Write $\mathcal{P}_{f_{1}}:=\\{p^{k}:f_{1}(p^{k})=0\\}$, so that
$f(p^{k})=F_{1}(p^{k})$ for all $p^{k}\notin\mathcal{P}_{f_{1}}$. Let $h(n)$
be the multiplicative function given at prime powers by
$h(p^{k}):=1-1_{\mathcal{P}_{f_{1}}}(p^{k})$. Combining Lemma 3.3 with Theorem
2.2, applied to the multiplicative function
$g=|f|_{\alpha}h=|F_{1}|_{\alpha}h,$ we deduce that there is $L=O_{\delta}(1)$
such that for each $\alpha\in X$ there is a primitive character
$\psi_{\alpha}$ of conductor $O_{\delta}(1)$ and a $t_{\alpha}\in[-Lx,Lx]$
such that

$\displaystyle\mathbb{D}(|f|_{\alpha},\psi_{\alpha}(n)n^{it_{\alpha}})^{2}$
$\displaystyle=\sum_{p\leq
x}\frac{1-\text{Re}(|f(p)|^{i\alpha}\overline{\psi}_{\alpha}(p)p^{-it_{\alpha}})}{p}$
$\displaystyle=\sum_{p\leq
x}\frac{1-\text{Re}(|F_{1}(p)|^{i\alpha}h(p)\overline{\psi}_{\alpha}(p)p^{-it_{\alpha}})}{p}+O(1)\ll_{\delta}1.$

Now let $\nu=O_{\delta}(1)$ be the maximum of the conductors of all the
$\psi_{\alpha}$, and set $m:=\nu!$. Then $\psi_{\alpha}^{m}(p)=1$ for all
$p\nmid m$. By the pretentious triangle inequality (8), uniformly over
$\alpha\in X$ we have

$\mathbb{D}(|f^{m}|^{i\alpha},n^{imt_{\alpha}};x)^{2}\leq
m^{2}\mathbb{D}(|f|_{\alpha},\psi_{\alpha}(n)n^{it_{\alpha}};x)^{2}+O_{\delta}(1)\ll_{\delta}1.$

Note that the above bound depended only on the validity of (15) and (11), both
of which are independent of $A$, as required. ∎

###  3.2. An application of the approximate Cauchy functional equation

We seek to understand the structure of the map $\beta\mapsto t_{\beta}$
arising in Corollary 3.4. In this direction we will use an idea of Halász [6],
improved upon in a paper by Ruzsa [6], which will allow us, roughly speaking,
to approximate this map by a linear map $\beta\mapsto c\beta$,
$c\in\mathbb{R}$, on some appropriate _sumset_ of $\tilde{X}:=m\cdot
X^{\prime}$. The proof is based on a few lemmas. Here and in the sequel, given
subsets $T_{1},T_{2}$ of an abelian group $(G,+)$, we define the sumset

$T_{1}+T_{2}:=\\{t_{1}+t_{2}:t_{1}\in T_{1},t_{2}\in T_{2}\\}\\\ $

and for $m\geq 2$ the iterated sumset

$mT:=\\{t_{1}+\cdots+t_{m}:t_{i}\in A\text{ for all }1\leq i\leq m\\}.$

In the sequel, for $S\subseteq\mathbb{R}$ and $\lambda>0$ we will also denote
by $\lambda\cdot S$ the dilation

$\lambda\cdot S:=\\{\lambda t:t\in S\\}.$

######  Lemma 3.5.

Let $D>0$ and let $Y\subseteq[-D,D]$ be a symmetric666We say that a set
$T\subseteq\mathbb{R}$ is _symmetric_ if $t\in T$ if and only if $-t\in T$.
set containing 0, and denote its Lebesgue measure by $\lambda(Y)$. Then if
$\ell\geq\left\lfloor 12D/\lambda(Y)\right\rfloor$ the sumset $\ell Y$
contains $[-D,D]$.

###### Proof.

Setting $Y^{\prime}:=\tfrac{1}{D}\cdot Y\subseteq[-1,1]$, we have
$\lambda(Y^{\prime})=\lambda(Y)/D$. Applying [15, Lemma (5.3)] to
$Y^{\prime}$, we get that $\ell Y^{\prime}$ contains $[-1,1]$ as soon as
$\ell\geq\left\lfloor 12/\lambda(Y^{\prime})\right\rfloor=\left\lfloor
12D/\lambda(Y)\right\rfloor$. The claim then follows, as
$y^{\prime}=y^{\prime}_{1}+\cdots+y^{\prime}_{\ell}$ holds with
$y^{\prime}_{j}\in Y^{\prime}$ if and only if
$Dy^{\prime}=y_{1}+\cdots+y_{\ell}$ with $y_{j}\in Y=D\cdot Y^{\prime}$. ∎

######  Lemma 3.6.

Let $K\geq 0$. Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be a bounded
function, and suppose that whenever
$\alpha_{1},\alpha_{2},\alpha_{1}+\alpha_{2}\in[-B,B]$ we have

$|\phi(\alpha_{1}+\alpha_{2})-\phi(\alpha_{1})-\phi(\alpha_{2})|\leq K.$

Then there is a $c\in\mathbb{R}$, specifically $c=\phi(B)/B$, with

$\sup_{\alpha\in[-B,B]}|\phi(\alpha)-c\alpha|\leq 3K.$

###### Proof.

Defining $\psi(\beta):=\phi(B\beta)/B$ for $\beta\in[-1,1]$, this follows
directly from applying [15, Lem. 5.11] to $\psi$ and rescaling by $B$. ∎

###### Proof of Proposition 3.1.

Let $\delta\in(0,1)$ and suppose $f:\mathbb{N}\rightarrow\mathbb{C}$ satisfies
(12) for $a,c\in\mathbb{N}$, $b,d\in\mathbb{Z}$ and $C\in\mathbb{C}$ and
$C(ad-bc)\neq 0$. Thus, there is a sequence $(x_{j})_{j}$ such that for $j\geq
1$ and $x=x_{j}$, (15) holds. Fix $j$ large for the moment. By Corollary 3.4,
there is a set $X\subseteq[-B,B]$ and an integer $m=O_{\delta}(1)$ such that
for each $\beta\in X$ there is $t_{\beta}=t_{\beta}(x_{j})$ with

(20) $\mathbb{D}(|f^{m}|_{\beta},n^{imt_{\beta}};x_{j})=O_{\delta}(1),$

where we recall that $f(p^{k})=F_{1}(p^{k})=f_{1}(p^{k})^{A}$ whenever
$f_{1}(p^{k})\neq 0$. The bound here is independent of $A$.
Lemma 3.5 shows that when $k=\lceil 100B/\lambda(X)\rceil\asymp_{\delta}1$ the
sumset $kX$ contains $[-B,B]$. In particular, every $\beta\in[-B,B]$ may be
written as

(21) $\beta=\alpha_{1}+\cdots+\alpha_{k},\quad\alpha_{\ell}\in X.$

Define $t_{\beta}:=t_{\alpha_{1}}+\cdots+t_{\alpha_{k}}$ in this case (if more
than one such decomposition exists, pick any of them). This defines a mapping
$\beta\mapsto t_{\beta}$ on all of $[-B,B]$, such that $|t_{\beta}|\leq
kALx_{j}<x_{j}^{3/2}$ uniformly, provided $x_{j}$ is sufficiently large with
respect to $\delta$. By the pretentious triangle inequality,

$\mathbb{D}(|f^{m}|_{\beta},n^{imt_{\beta}};x_{j})=\mathbb{D}(|f^{m}|^{i(\alpha_{1}+\cdots+\alpha_{k})},n^{im(t_{\alpha_{1}}+\cdots+t_{\alpha_{k}})};x_{j})\leq\sum_{\ell=1}^{k}\mathbb{D}(|f^{m}|_{\alpha_{\ell}},n^{imt_{\alpha_{\ell}}};x_{j})\ll_{\delta}1.$

Moreover, if $\beta_{1},\beta_{2},\beta_{1}+\beta_{2}\in[-B,B]$ then

$\displaystyle\mathbb{D}(n^{imt_{\beta_{1}+\beta_{2}}},n^{im(t_{\beta_{1}}+t_{\beta_{2}})};x_{j})$
$\displaystyle\leq\mathbb{D}(|f^{m}|_{\beta_{1}+\beta_{2}},n^{imt_{\beta_{1}+\beta_{2}}};x_{j})+\mathbb{D}(|f^{m}|_{\beta_{1}}n^{imt_{\beta_{1}}};x_{j})+\mathbb{D}(|f^{m}|_{\beta_{2}},n^{imt_{\beta_{2}}};x_{j})$
$\displaystyle\ll_{\delta}1.$

Since
$m|t_{\beta_{1}+\beta_{2}}-t_{\beta_{1}}-t_{\beta_{2}}|=O_{\delta}(x_{j}^{3/2})<x_{j}^{2}$
for $j$ sufficiently large, Lemma 2.1 yields

$\log(1+m|t_{\beta_{1}+\beta_{2}}-t_{\beta_{1}}-t_{\beta_{2}}|\log
x_{j})\ll_{\delta}1.$

It follows that there is $K=O_{\delta}(1)$ such that for any
$\beta_{1},\beta_{2}\in[-B,B]$ with $\beta_{1}+\beta_{2}\in[-B,B]$,

$|t_{\beta_{1}+\beta_{2}}-t_{\beta_{1}}-t_{\beta_{2}}|\leq\frac{K}{\log
x_{j}}.$

By Lemma 3.6 there is $r_{j}=r_{j}(A)\in\mathbb{R}$ such that uniformly over
$\beta\in[-B,B]$,

$t_{\beta}=r_{j}\beta+O_{\delta}\left(\frac{1}{\log x_{j}}\right);$

since $|t_{\beta}|<x_{j}^{3/2}$, say, for $j$ large enough, we see that
$|r_{j}|<x_{j}^{2}$. It follows that

$p^{im(t_{\beta}-r_{j}\beta)}=1+O_{\delta}\left(\frac{mK\log p}{\log
x_{j}}\right)\text{ for all }p\leq x_{j}^{1/(3mK)}.$

Applying (8), we find that, uniformly over $\beta\in[-B,B]$,

$\displaystyle\mathbb{D}(|f^{m}|_{\beta},n^{imr_{j}\beta};x_{j})^{2}$
$\displaystyle=\sum_{p\leq
x_{j}^{1/(3mK)}}\frac{1-\text{Re}(|f(p)|^{im\beta}p^{-imt_{\beta}})}{p}+O\left(\sum_{x_{j}^{1/(3mK)}<p\leq
x_{j}}\frac{1}{p}+\frac{mK}{\log x_{j}}\sum_{p\leq x_{j}^{1/(3mK)}}\frac{\log
p}{p}\right)\ll_{\delta}1.$

Equivalently, we may write this as

(22) $\max_{\beta\in[-B,B]}\sum_{p\leq
x_{j}}\frac{1-\text{Re}((|f(p)|/p^{r_{j}})^{im\beta})}{p}\ll_{\delta}1,$

emphasising once again that this bound is independent of $A$. We may now
complete the proof of the proposition. Given (22), after the change of
variables $\beta\mapsto m\beta$ we observe that

$\frac{1}{2mB}\int_{-mB}^{mB}\sum_{p\leq
x_{j}}\frac{1-\text{Re}((|f(p)|/p^{r_{j}})^{i\beta})}{p}d\beta=O_{\delta}(1).$

Since $\frac{1}{2R}\int_{-R}^{R}y^{it}dt=\frac{\sin(R\log y)}{R\log
y}=:\text{sinc}(R\log y)$, we deduce that

$\sum_{p\leq
x_{j}}\frac{1-\text{sinc}(mB\log(|f(p)|/p^{r_{j}}))}{p}=O_{\delta}(1).$

As $1-\text{sinc}(t)\gg 1$ whenever $|t|\geq 2$, recalling the notation of the
proposition and that $f(p)\neq f_{1}(p)$ on a sparse set, we deduce using
$m\geq 1$ that

$\sum_{\begin{subarray}{c}p\leq x_{j}\\\
|F_{1}(p)|/p^{r_{j}}\notin[e^{-2/B},e^{2/B}]\end{subarray}}\frac{1}{p}\leq\sum_{\begin{subarray}{c}p\leq
X\\\ |\log(|f(p)|/p^{r_{j}})|\geq
2/(mB)\end{subarray}}\frac{1}{p}+O(1)=O_{\delta}(1).$

Recalling that $F_{1}=f_{1}^{A}$, the claim follows. ∎

##  4\. Proof of the Converse Result

###### Proof of Proposition 1.5.

Let $f:\mathbb{N}\rightarrow\mathbb{R}\backslash\\{0\\}$ be multiplicative,
satisfying the condition

(23) $\sum_{p:|f(p)|\neq 1}\frac{1}{p}<\infty.$

Observe that if there is $c_{f}^{\prime}>0$ such that

(24) $\frac{1}{\log x}\sum_{n\leq x}\frac{1_{|f(n)|=|f(n+2)|}}{n}\geq
c_{f}^{\prime}+o(1),$

then by the pigeonhole principle there is a $u_{x}\in\\{-1,+1\\}$ such that if
$c_{f}:=c_{f}^{\prime}/2$ then

$\frac{1}{\log x}\sum_{n\leq x}\frac{1_{f(n)=u_{x}f(n+2)}}{n}\geq c_{f}+o(1).$

Pigeonholing once again, we can choose an infinite sequence $(x_{j})_{j}$ on
which $u=u_{x_{j}}\in\\{-1,+1\\}$ is constant, in which case we obtain

$\limsup_{x\rightarrow\infty}\frac{1}{\log x}\sum_{n\leq
x}\frac{1_{f(n)=uf(n+2)}}{n}>0,$

as claimed.
Moreover, the condition (23) is unchanged when passing from $f$ to $|f|$.
Thus, in the sequel, we replace $f$ by $|f|$ and assume that $f\geq 0$. We
will show that (24) holds whenever $x\geq x_{0}(f)$.
By partial summation, it will suffice to show that

$\sum_{n\leq x}1_{f(n)=f(n+2)}\geq(c_{f}^{\prime}+o(1))x$

for all $x\geq x_{0}(f)$. To this end, we restrict to $n\equiv 1\pmod{2}$, so
that $n$ and $n+2$ are coprime, and also such that $n,n+2$ are both
squarefree. Observe that for such $n$, if $f(p)=1$ for all $p||n$ and all
$p||n+2$ then $f(n)=1=f(n+2)$. Thus, we obtain

(25) $\sum_{n\leq x}1_{f(n)=f(n+2)}\geq\sum_{\begin{subarray}{c}n\leq x\\\
p|n(n+2)\Rightarrow f(p)=1\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid n}.$

Take $z:=(\log x)^{0.99}$. As $\sum_{p:f(p)\neq 1}p^{-1}<\infty$, we find

$\sum_{\begin{subarray}{c}n\leq x\\\ \exists p>z:\\\ p|n(n+2),\,f(p)\neq
1\end{subarray}}1\ll\sum_{\begin{subarray}{c}z<p\leq x\\\ f(p)\neq
1\end{subarray}}\left(\frac{2x}{p}+1\right)\ll
x\left(\sum_{\begin{subarray}{c}p>z\\\ f(p)\neq
1\end{subarray}}\frac{1}{p}+\frac{1}{\log x}\right)=o(x).$

It follows that

$\sum_{\begin{subarray}{c}n\leq x\\\ p|n(n+2)\Rightarrow
f(p)=1\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid
n}=\sum_{\begin{subarray}{c}n\leq x\\\ p|n(n+2),\,p\leq z\Rightarrow
f(p)=1\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid n}+o(x).$

If we set

$P(z):=\prod_{\begin{subarray}{c}3\leq p\leq z\\\ f(p)\neq 1\end{subarray}}p,$

then the prime number theorem trivially gives $P(z)\leq x^{0.01}$ for large
enough $x$. Thus, for any divisors $d_{1},d_{2}|P(z)$ we have $d_{1}d_{2}\leq
x^{0.02}$. Since $2\nmid P(z)$, by Möbius inversion we obtain

$\displaystyle\sum_{\begin{subarray}{c}n\leq x\\\ p|n(n+2),\,p\leq
z\Rightarrow f(p)=1\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid n}$
$\displaystyle=\sum_{\begin{subarray}{c}d_{1},d_{2}|P(z)\\\
(d_{1},d_{2})=1\end{subarray}}\mu(d_{1})\mu(d_{2})\sum_{\begin{subarray}{c}n\leq
x\\\ n\equiv 0\pmod{d_{1}}\\\ n+2\equiv
0\pmod{d_{2}}\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid n}.$

By e.g. [12, Lem. 3.3] (taking $a=2$) the inner sum in this last expression
is, for each pair of coprime divisors $d_{1},d_{2}$ of $P(z)$,

$x\frac{\phi(d_{1})\phi(d_{2})}{2(d_{1}d_{2})^{2}}\prod_{p\nmid
d_{1}d_{2}}(1-\frac{2}{p^{2}})+O_{\varepsilon}(x^{2/3+\varepsilon}).$

Summing over all such $d_{1},d_{2}$, we thus obtain

$Cx\sum_{\begin{subarray}{c}d_{1},d_{2}|P(z)\\\
(d_{1},d_{2})=1\end{subarray}}\frac{\mu(d_{1})\mu(d_{2})\phi(d_{1})\phi(d_{2})}{(d_{1}d_{2})^{2}}\prod_{p|d_{1}d_{2}}\left(1-\frac{2}{p^{2}}\right)^{-1}+O(x^{0.7}),$

where $C:=\frac{1}{2}\prod_{p}(1-2p^{-2})$. As

$\sum_{\begin{subarray}{c}d\geq 1\\\ p|d\Rightarrow f(p)\neq
1\end{subarray}}\frac{1}{d}\ll\exp\left(\sum_{p:f(p)\neq
1}\frac{1}{p}\right)\ll 1,$

due to the trivial bound $\phi(d_{1})\leq d_{1}$ the double series in
$d_{1},d_{2}|P(z)$ converges absolutely as $z\rightarrow\infty$, and therefore

$\displaystyle C\sum_{\begin{subarray}{c}d_{1},d_{2}|P(z)\\\
(d_{1},d_{2})=1\end{subarray}}\frac{\mu(d_{1})\mu(d_{2})\phi(d_{1})\phi(d_{2})}{(d_{1}d_{2})^{2}}\prod_{p|d_{1}d_{2}}\left(1-\frac{2}{p^{2}}\right)^{-1}$
$\displaystyle=C\sum_{\begin{subarray}{c}d_{1},d_{2}\geq 1\\\
p|d_{1}d_{2}\Rightarrow f(p)\neq 1\\\ (d_{1},d_{2})=1\\\ 2\nmid
d_{1}d_{2}\end{subarray}}\frac{\mu(d_{1})\mu(d_{2})\phi(d_{1})\phi(d_{2})}{(d_{1}d_{2})^{2}}\prod_{p|d_{1}d_{2}}\left(1-\frac{2}{p^{2}}\right)^{-1}+o(1).$

Upon computing Euler products we find the asymptotic estimate

$\sum_{\begin{subarray}{c}n\leq x\\\ p|n(n+2)\Rightarrow
f(p)=1\end{subarray}}\mu^{2}(n)\mu^{2}(n+2)1_{2\nmid
n}=(c_{f}^{\prime}+o(1))x,$

where $c_{f}^{\prime}$ is defined by the (convergent) product

$c_{f}^{\prime}:=\prod_{\begin{subarray}{c}p\geq 3\\\ f(p)\neq
1\end{subarray}}\left(1-\frac{2(p-1)}{p^{2}-2}\right)>0.$

Combined with (25), this leads to the lower bound

$\sum_{n\leq x}1_{f(n)=f(n+2)\neq 0}\geq(c_{f}^{\prime}+o(1))x,$

as claimed.

∎

##  5\. Proof of Corollary 1.6

Let $\phi$ be a holomorphic cuspidal eigenform without CM for
$\text{SL}_{2}(\mathbb{Z})$, of weight $k\geq 2$, normalised so its sequence
of Fourier coefficients $(a_{\phi}(n))_{n}$ satisfies $a_{\phi}(1)=1$. It is
well-known that the map $n\mapsto a_{\phi}(n)$ is multiplicative, so that
$f=a_{\phi}$ fits the setup of Theorem 1.1.

To prove Corollary 1.6 it suffices to check that (1) holds for $a_{\phi}(n)$,
after which the claim then follows from Theorem 1.1 (applied to each of the
equations $a_{\phi}(n)=a_{\phi}(n+j)$, $1\leq|j|\leq h$).

Set $\lambda_{\phi}(n):=a_{\phi}(n)/n^{(k-1)/2}$. By the prime number theorem
for Rankin-Selberg $L$-functions (e.g. combining [10, Cor. 1.2 and Lem. 5.1]
with [7, Exer. 6]),

(26) $\sum_{p\leq X}|\lambda_{\phi}(p)|^{2}\log p=X+O(Xe^{-c\sqrt{\log X}}),$

for some $c=c_{\phi}>0$. We claim that if $y_{0}$ is sufficiently large and
$y\geq y_{0}$ then

(27) $\sum_{\begin{subarray}{c}y\leq p\leq 2y\\\ |a_{\phi}(p)|\neq
1\end{subarray}}\log p\geq y/5.$

Indeed, suppose (27) fails for some $y\geq y_{0}$, with $y_{0}$ large enough.
From (26) and Deligne’s bound $|a_{\phi}(p)|\leq 2p^{(k-1)/2}$ we obtain

$\sum_{\begin{subarray}{c}y\leq p\leq 2y\\\
|a_{\phi}(p)|=1\end{subarray}}|a_{\phi}(p)p^{-(k-1)/2}|^{2}\log p\geq
9y/10-4\sum_{\begin{subarray}{c}y\leq p\leq 2y\\\ |a_{\phi}(p)|\neq
1\end{subarray}}\log p>y/10.$

But by the prime number theorem, the left-hand side of this last expression is

$\leq y^{1-k}\sum_{\begin{subarray}{c}y\leq p\leq 2y\end{subarray}}\log p\ll
y^{2-k},$

which is impossible for $y_{0}$ large enough and $k\geq 2$.

By a dyadic decomposition and (27), we find

$\sum_{\begin{subarray}{c}p\leq x\\\ |a_{\phi}(p)|\neq
1\end{subarray}}\frac{1}{p}\geq\sum_{y_{0}\leq 2^{j}\leq
x/2}\frac{1}{2^{j+1}\log(2^{j+1})}\sum_{\begin{subarray}{c}2^{j}\leq p\leq
2^{j+1}\\\ |a_{\phi}(p)|\neq 1\end{subarray}}\log p\geq\frac{1}{10\log
2}\sum_{y_{0}\leq 2^{j}\leq x/2}\frac{1}{j+1}=c\log\log x+O(1),$

with $c=1/(10\log 2)$, so (1) clearly holds as well.

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