# Freezing transition and moments of moments of the Riemann zeta function

Michael J. Curran  Mathematical Institute, University of Oxford, Oxford, OX2
6GG, United Kingdom.

###### Abstract.

Moments of moments of the Riemann zeta function, defined by

$\text{MoM}_{T}(k,\beta):=\frac{1}{T}\int_{T}^{2T}\Bigg{(}\int\displaylimits_{|h|\leq(\log
T)^{\theta}}|\zeta(\tfrac{1}{2}+it+ih)|^{2\beta}~{}dh\Bigg{)}^{k}~{}dt$

where $k,\beta\geq 0$ and $\theta>-1$, were introduced by Fyodorov and Keating
[21] when comparing extreme values of zeta in short intervals to those of
characteristic polynomials of random unitary matrices. We study the $k=2$ case
as $T\rightarrow\infty$ and obtain sharp upper bounds for
$\text{MoM}_{T}(2,\beta)$ for all real $0\leq\beta\leq 1$ as well as lower
bounds of the conjectured order for all $\beta\geq 0$. In particular, we show
that the second moment of moments undergoes a freezing phase transition with
critical exponent $\beta=\tfrac{1}{\sqrt{2}}$.

##  1\. Introduction

Moments of the Riemann zeta function and other $L$-functions are classical
objects in analytic number theory utilized in the study large values and the
value distribution of $L$-functions. It has long been conjectured that for
$\beta\geq 0$ there is an asymptotic

$\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it)|^{2\beta}\sim C_{\beta}(\log
T)^{\beta^{2}}$

as $T\rightarrow\infty$ for certain constants $C_{\beta}$. The values of the
constants $C_{\beta}$ were conjectured recently in the work of Keating and
Snaith [32] using analogies between the Riemann zeta function and
characteristic polynomials of random matrices. These asymptotics have only
been established in the case of $\beta=1$ due to work of Hardy and Littlewood
[25] and the case of $\beta=2$ in work of Ingham [23], however.

While relatively little is known about the moments of zeta for $\beta\geq 3$,
in recent literature there has been interest in moments and the value
distribution of zeta in short intervals on the critical line. Instead of
studying zeta along an interval $[T,2T]$, one instead chooses a point
$t\in[T,2T]$ uniformly at random and then tries to understand the typical
behavior of the moments or maximum of zeta in the shorter interval
$[t-1,t+1]$, say. The question of the maximum of zeta in short intervals is
the subject of the Fyodorov-Hiary-Keating conjecture [20, 21], which predicts
that if $t$ is chosen uniformly at random from $[T,2T]$ then as
$T\rightarrow\infty$

$\mathbb{P}\left(\max_{|h-t|\leq 1}|\zeta(\tfrac{1}{2}+it+ih)|>e^{y}\frac{\log
T}{(\log\log T)^{3/4}}\right)\rightarrow 1-F(y),$ (1)

where $F(y)$ is a cumulative distribution function with tail decay $1-F(y)\sim
ye^{-2y}$ as $y\rightarrow\infty$. The fraction $\frac{3}{4}$ and the tail
decay are manifestations of correlations of nearby shifts of zeta. If these
correlations were not present, probabilistic heuristics would predict a
$\tfrac{1}{4}$ in place of $\tfrac{3}{4}$ and a tail decay of $e^{-2y}$
instead of $ye^{-2y}$. An analogous conjecture is known to hold for the
circular-$\beta$ ensemble in the random matrix setting– refer to the work
Paquette and Zeitouni [38, 39].

There has been a lot of recent progress toward the conjecture (1).
Conditionally on the Riemann hypothesis, Najnudel showed with probability
$1-o_{\varepsilon}(1)$ that $\max_{|h-t|\leq 1}|\zeta(\tfrac{1}{2}+it+ih)|$ is
bounded by $(\log T)^{1+\varepsilon}$, as well corresponding upper and lower
bounds for the imaginary part of $\log\zeta$ [37]. Unconditionally Arguin,
Belius, Bourgade, Radziwiłł, and Soundararajan [1] proved that
$\max_{|h-t|\leq 1}|\zeta(\tfrac{1}{2}+it+ih)|$ lies in the interval $[(\log
T)^{1-\varepsilon},(\log T)^{1+\varepsilon}]$ with probability
$1-o_{\varepsilon}(1)$. Harper [26] then refined the upper bound showing that
with probability $1-o(1)$

$\max_{|h-t|\leq 1}|\zeta(\tfrac{1}{2}+it+ih)|\leq\frac{\log T}{(\log\log
T)^{3/4+o(1)}}.$

Most recently Arguin, Bourgade, and Radziwiłł established the upper bound in
the Fyodorov-Hiary-Keating conjecture [2]. Indeed, one consequence of [2] is
that for fixed $y$ the probability in (1) is $O(ye^{-2y})$ as
$T\rightarrow\infty$.

Moments of zeta in short intervals were also recently studied Fyodorov and
Keating [21] followed by Arguin, Ouimet, and Radziwiłł [3]. The problem now is
to understand the behavior of the short moments

$\int_{|h|\leq(\log T)^{\theta}}|\zeta(\tfrac{1}{2}+it+ih)|^{2\beta}dt$ (2)

where $\theta>-1$ and $t$ is chosen uniformly at random from $[T,2T]$. The
typical behavior of these short moments was investigated by Fyodorov and
Keating [21] when $\theta=0$, and their behavior for all $\theta>-1$ was
determined by Arguin, Ouimet, and Radziwiłł [3]. One particularly interesting
conclusion of the analysis is that there is a so-called freezing transition as
one varies the parameter $\beta$. More precisely, for fixed $\theta>-1$ there
is a critical value $\beta_{c}(\theta)$ such that when
$\beta<\beta_{c}(\theta)$, the moments (2) are governed by the typical values
of $\zeta(\tfrac{1}{2}+it)$, yet when $\beta>\beta_{c}(\theta)$ the moments
(2) are dominated by just a few large values of $\zeta(\tfrac{1}{2}+it)$.

In this paper, we will not be interested in the typical values of the moments
(2), but instead larger values. More precisely, we will investigate the $k=2$
case of the so-called _moments of moments_ of zeta:

$\text{MoM}_{T}(k,\beta):=\frac{1}{T}\int_{T}^{2T}\Bigg{(}~{}\int\displaylimits_{|h|\leq(\log
T)^{\theta}}|\zeta(\tfrac{1}{2}+it+ih)|^{2\beta}~{}dh\Bigg{)}^{k}~{}dt$

where $\beta\geq 0$ and $\theta>-1$. In other terms, we study the variance of
the moments (2). The problem of moments of moments was first studied by
Fyodorov, Hiary, and Keating in the random matrix setting [20, 21], who
conjectured asymptotics for

$\text{MoM}_{U(N)}(k,\beta):=\int_{U(N)}\left(\frac{1}{2\pi}\int_{0}^{2\pi}|P_{N}(A,\theta)|^{2\beta}d\theta\right)^{k}dA$

as $N\rightarrow\infty$ where $dA$ is the Haar measure on the unitary group
$U(N)$ of size $N$ and $P_{N}(A,\theta):=\det(I-Ae^{-i\theta})$ is the
characteristic polynomial of $A$. There has been a lot of recent progress on
these conjectures, and it has now been proven that (see [4, 5, 12, 19]) for
$\beta\geq 0$ and $k\in\mathbb{N}$ that

$\text{MoM}_{U(N)}(k,\beta)\sim\begin{cases}A(k,\beta)N^{k\beta^{2}}&k<1/\beta^{2}\\\
B(k,\beta)N^{k^{2}\beta^{2}-k+1}&k>1/\beta^{2}\end{cases}$

for certain constants $A(k,\beta),B(k,\beta)$ as $N\rightarrow\infty$. For
$k\geq 2$, the transition at the critical exponent $\beta=1/\sqrt{k}$ was
analyzed by Keating and Wong [34], who showed

$\text{MoM}_{U(N)}(1/\beta^{2},\beta)\sim D_{\beta}N\log N$

for certain explicit constants $D_{\beta}$ that also appear in the theory of
Gaussian multiplicative chaos.

In the case $\theta=0$, the random matrix philosophy of Keating and Snaith
[32] leads us to the prediction that if $k\geq 2$ is an integer and $\beta\geq
0$ then

$\text{MoM}_{T}(k,\beta)\sim\begin{cases}A^{\prime}(k,\beta)(\log
T)^{k\beta^{2}}&k<1/\beta^{2}\\\ B^{\prime}(k,\beta)(\log T)(\log\log
T)&k=1/\beta^{2}\\\ C^{\prime}(k,\beta)(\log
T)^{k^{2}\beta^{2}-k+1}&k>1/\beta^{2}\end{cases}$

for certain constants
$A^{\prime}(k,\beta),B^{\prime}(k,\beta),C^{\prime}(k,\beta)$ as
$T\rightarrow\infty$. These conjectures were also shown to hold for
$\beta,k\in\mathbb{N}$ by Bailey and Keating [6] assuming the standard
conjectures on shifted moments of zeta due to Conrey, Farmer, Keating,
Rubinstein, and Snaith [16]. These asymptotics do not however include the
critical or subcritical regimes where $\beta\leq 1/\sqrt{k}$.

The goal of this paper is to provide upper and lower bounds of the expected
order of magnitude in the case $k=2$ and non-integer $\beta$. In particular we
will show that there is indeed a phase transition at
$\beta=\tfrac{1}{\sqrt{2}}$, although this transition occurs in the sub
leading order terms when $\theta>0$. The starting point of the method is to
rewrite the integral as

$\text{MoM}_{T}(2,\beta)=\frac{1}{T}\iint\displaylimits_{|h_{1}|,|h_{2}|\leq(\log
T)^{\theta}}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}~{}dt~{}dh_{1}~{}dh_{2}.$
(3)

Therefore we can understand the $\text{MoM}_{T}(2,\beta)$ by understanding the
correlation structure of zeta in short intervals and then integrating over all
of the possible shifts $h_{1},h_{2}$. Morally, we will find that if
$|h_{1}-h_{2}|\ll 1/\log T$ then

$\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}~{}dt\approx(\log
T)^{4\beta^{2}},$

while when $|h_{1}-h_{2}|\gg 1/\log T$ that

$\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}~{}dt\approx(\log
T)^{2\beta^{2}}|\zeta(1+i(h_{1}-h_{2}))|^{2\beta^{2}}.$

So $\zeta(\tfrac{1}{2}+it+ih_{1})$ and $\zeta(\tfrac{1}{2}+it+ih_{2})$ are
almost perfectly correlated when $h_{1}$ and $h_{2}$ are close, and they
decorrelate for more distant shifts. Indeed one may see that the factor of
$|\zeta(1+i(h_{1}-h_{2}))|^{2\beta^{2}}$ is bounded on average by computing
moments of zeta on the one line. A similar correlation structure for
$\log|\zeta(\tfrac{1}{2}+it)|$ was proven by Bourgade [11], and this
correlation structure is what gives rise to the phase transition at
$\beta=1/\sqrt{2}$. However as we look at larger shifts, we also see that a
large value of $\zeta(1+ih)$ on the one line will cause
$\zeta(\tfrac{1}{2}+it)$ and $\zeta(\tfrac{1}{2}+it+ih)$ to have an unusually
large average correlation.

For $\beta=1$, one can obtain an asymptotic formula for $\text{MoM}_{T}(2,1)$
as $T\rightarrow\infty$ by using the work of Bettin on the shifted second
moment of zeta with unbounded shifts [8] for example. Similar techniques can
also establish an asymptotic for $\text{MoM}(2,2)$ unconditionally– one would
just need to work a bit to obtain uniformity over shifts of size $(\log
T)^{\theta}$. In fact, asymptotics for $\text{MoM}_{T}(2,1)$ and
$\text{MoM}_{T}(2,2)$ are straightforward consequences of Theorems 2.2 and 2.4
respectively– refer to section 2. However, both of these asymptotics do not
address the particularly interesting critical case $\beta=1/\sqrt{2}$. To
handle non-integer values of $\beta$, we use a principal pioneered in [29] and
[42]: if one can asymptotically evaluate the twisted $2k^{\text{th}}$ moment
of a family of $L$-functions for some $k>0$, then one can obtain upper bounds
of the correct order for the $2\beta^{\text{th}}$ moment in that family when
$\beta\leq k$ as well as sharp lower bounds for for all $\beta\geq 0$. This
method has been used to give conjecturally sharp upper and lower bounds for
moments of several families of $L$-functions; see [18, 29, 30, 41, 42] for
example.

The first task of this paper will be to establish the corresponding twisted
second and fourth moment formulae with shifts of order $(\log T)^{\theta}$.
The second moment can be handled using a slight modification of the techniques
used in Young’s proof of Levinson’s theorem [45], and the fourth moment will
follow from some minor modifications to the work of Hughes and Young on the
twisted fourth moment [31]. With these formulae in hand, we may then follow
the principal of [29] and [42] to establish sharp upper bounds for
$\text{MoM}_{T}(2,\beta)$ for all $\beta\leq 1$, and lower bounds for all
$\beta\geq 0$.

######  Theorem 1.1.

Let $-1<\theta\leq 0$. If $0\leq\beta<1/\sqrt{2}$ then

$\text{MoM}_{T}(2,\beta)\ll\frac{1}{1-2\beta^{2}}(\log
T)^{2\beta^{2}(1+\theta)}.$

For $1/\sqrt{2}<\beta\leq 1$

$\text{MoM}_{T}(2,\beta)\ll\frac{1}{2\beta^{2}-1}(\log
T)^{4\beta^{2}+\theta-1},$

and for $\beta=1/\sqrt{2}$

$\text{MoM}_{T}(2,\tfrac{1}{\sqrt{2}})\ll(1+\theta)(\log
T)^{1+\theta}(\log\log T).$

Instead if $\theta>0$, then if $0\leq\beta\leq\min(\sqrt{(1+\theta)/2},1)$

$\text{MoM}_{T}(2,\beta)\ll(\log
T)^{2\beta^{2}+2\theta}+\frac{1}{1-2\beta^{2}}(\log T)^{2\beta^{2}+\theta},$

where the sub leading term is replaced by $(1+\theta)(\log
T)^{1+\theta}(\log\log T)$ if $\beta=1/\sqrt{2}$. In the case
$\min((\sqrt{1+\theta})/2,1)\leq\beta\leq 1$ we have

$\text{MoM}_{T}(2,\beta)\ll\frac{1}{2\beta^{2}-1}(\log
T)^{4\beta^{2}-1+\theta}.$

All implied constants are absolute assuming $T$ is taken sufficiently large in
terms of $\theta$.

######  Remark.

For $\theta>0$ and $0\leq\beta\leq\min(\sqrt{(1+\theta)/2},1)$, the sub
leading order term can be removed provided one that $T$ is sufficiently large
in terms of $\beta$ or if one allows the implicit constant to depend on
$\beta$. This technicality is necessary because the phase transition occurs in
the sub leading order terms of the second moment of moments.

######  Theorem 1.2.

Let $-1<\theta\leq 0$. If $0\leq\beta<1/\sqrt{2}$ then

$\text{MoM}_{T}(2,\beta)\gg_{\beta}(\log T)^{2\beta^{2}(1+\theta)}.$

For $\beta\geq 1/\sqrt{2}$

$\text{MoM}_{T}(2,\beta)\gg_{\beta}(\log T)^{4\beta^{2}+\theta-1},$

and for $\beta=1/\sqrt{2}$

$\text{MoM}_{T}(2,\tfrac{1}{\sqrt{2}})\gg(1+\theta)(\log
T)^{1+\theta}(\log\log T).$

Instead if $\theta>0$, then if $0\leq\beta\leq\sqrt{(1+\theta)/2}$

$\text{MoM}_{T}(2,\beta)\gg_{\beta}(\log T)^{2\beta^{2}+2\theta}.$

When $\beta\geq\sqrt{(1+\theta)/2}$, we have

$\text{MoM}_{T}(2,\beta)\gg_{\beta}\frac{1}{2\beta^{2}-1}(\log
T)^{4\beta^{2}-1+\theta},$

where all implied constants depend only on $\beta$ provided $T$ is large
enough in terms of $\theta$.

We will first establish the necessary twisted moment formulae in section 2.
Subsequently we will prove Theorem 1.1 in section 3 and then Theorem 1.2 in
section 4. We now introduce some notation we will use throughout the paper.
The notation is very similar to that of [29], with only a few cosmetic
differences. Given some positive integer $\ell$ and parameters
$T_{0}<T_{1}<\ldots<T_{\ell}$, define for $1\leq j\leq\ell$

$\mathcal{P}_{j}(s):=\textup{Re }\sum_{T_{j-1}<p\leq T_{j}}p^{-s},\qquad
P_{j}:=\sum_{T_{j-1}<p\leq T_{j}}p^{-1}.$

If $T_{\ell}$ is chosen to be some small power of $T$, then the sum of all the
$\mathcal{P}_{j}(s)$ will be a good proxy for $\log|\zeta(s)|$ on average [29,
30], and each $P_{j}$ can be thought of as the variance of each
$\mathcal{P}_{j}(s)$ on the half line. Then given parameters $K_{j}\geq 0$ for
$1\leq j\leq\ell$ let

$\mathcal{N}_{j}(s;\beta):=\sum_{m=0}^{K_{j}}\frac{1}{m!}(\beta\mathcal{P}_{j}(s))^{m}=\sum_{\begin{subarray}{c}p\mid
n\Rightarrow p\in(T_{j-1},T_{j}]\\\ \Omega(n)\leq
K_{j}\end{subarray}}\frac{\beta^{\Omega(n)}g(n)}{n^{s}}$

where $g(n)$ is the multiplicative function such that $g(p^{a})=1/a!$ and
$\Omega(n)$ is the number of prime divisors of $n$ counting multiplicity.
Notice that $\mathcal{N}_{j}(s;\beta)$ is simply a truncation of the series
expansion for $\exp(\beta\mathcal{P}_{j}(s))$. Since $\mathcal{P}_{j}(s)$ is a
reasonable proxy for $\log|\zeta(s)|$, if we choose parameters $K_{j}$ to be
large multiples of the variance $P_{j}$ then one might expect

$\mathcal{N}(s;\beta):=\prod_{j\leq\ell}\mathcal{N}_{j}(s;\beta)$

to behave like $\zeta(s)^{\beta}$ on average on the half line. The main
difficulty here is that if the $\mathcal{P}_{j}(s)$ are unusually large for
some bad set of $s$, then the series approximation for
$\exp(\beta\mathcal{P}_{j}(s))$ will be weaker. Fortunately, such events are
rare, and we can discard the contribution of such bad $s$ using the
incremental structure of $\mathcal{N}(s;\beta)$.

## Acknowledgements

The author would like to thank his supervisor Jon Keating for suggesting this
problem and for helpful discussions and feedback.

##  2\. Twisted moments with mesoscopic shifts

###  2.1. Twisted second moment

An asymptotic for the twisted second moment of zeta with unbounded shifts is
not necessary to prove Theorems 1.1 or 1.2. However, such an estimate will
imply an asymptotic for $\text{MoM}_{T}(2,1)$ and will serve as a good warm up
to the twisted fourth moment estimate we will require.

Before stating our twisted moment estimate, we introduce some convenient
notation. Denote

$\lambda(s)=\pi^{s-1/2}\frac{\Gamma((1-s)/2)}{\Gamma(s/2)}.$

Therefore the functional equation for $\zeta$ can be expressed as
$\zeta(s)=\lambda(1-s)\zeta(1-s)$, and $\lambda$ satisfies its own functional
equation $\lambda(s)\lambda(1-s)=1$. Define

$G(s)=e^{s^{2}}\left(1-\left(\frac{2s}{\alpha_{1}+\alpha_{2}}\right)^{2}\right),$
$g_{\alpha_{1},\alpha_{2}}(s,t)=\pi^{-s}\frac{\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{1}+s+it}{2}\right)\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{2}+s-it}{2}\right)}{\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{1}+it}{2}\right)\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{2}-it}{2}\right)},$
$X_{\alpha_{1},\alpha_{2},t}=\lambda(\tfrac{1}{2}+\alpha_{1}+it)\lambda(\tfrac{1}{2}+\alpha_{2}-it).$

The precise definition of $G$ here is not too important– the key properties
are that $G(0)=1$, $G$ has a zero at $\tfrac{\alpha_{1}+\alpha_{2}}{2}$, and
$G$ decays rapidly in any vertical strip. By equation 4.12.3 of [44], we can
approximate

$\displaystyle X_{\alpha_{1},\alpha_{2},t}=$
$\displaystyle\left(1+O\left(\frac{1}{t}\right)\right)$
$\displaystyle\times\exp\Bigg{(}(\alpha_{1}+it)\log\left(\frac{2\pi}{t+\textup{Im
}\alpha_{1}}\right)$
$\displaystyle+(\alpha_{2}-it)\log\left(\frac{2\pi}{t-\textup{Im
}\alpha_{2}}\right)+\alpha_{1}-\alpha_{2}\Bigg{)}.$

Therefore if we define

$Y_{\alpha_{1},\alpha_{2},t}:=\exp\left(\frac{\alpha_{1}+it}{2}\log\left(\frac{2\pi}{t+\textup{Im
}\alpha_{1}}\right)+i\frac{\alpha_{2}-it}{2}\log\left(\frac{2\pi}{t-\textup{Im
}\alpha_{2}}\right)+\frac{\alpha_{1}-\alpha_{2}}{2}\right)$

we have the approximation
$X_{\alpha_{1},\alpha_{2}}=Y_{\alpha_{1},\alpha_{2}}^{2}(1+O(1/t))$. We will
use this quantity to symmetrize our formula for the twisted second moment.
Finally, we write

$V_{\alpha_{1},\alpha_{2}}(x,t)=\frac{1}{2\pi
i}\int_{(1)}\frac{G(s)}{s}g_{\alpha_{1},\alpha_{2}}(s,t)x^{-s}~{}ds.$

By Proposition 5.4 of [24] it follows that for $A\geq 0$,
$j\in\mathbb{Z}_{\geq 0}$ and $T$ sufficiently large in terms of $\theta$

$t^{j}\frac{\partial^{j}}{\partial
t^{j}}V_{\alpha_{1},\alpha_{2}}(x,t)\ll_{A,j}(1+|x|/t)^{-A}.$ (4)

We will make use of the following version of the approximate functional
equation given by theorem 5.3 of [24].

######  Lemma 2.1.

Let $t\in[T/4,4T]$, $\textup{Re }\alpha_{j}\ll 1/\log T$, $|\textup{Im
}\alpha_{j}|\leq(\log T)^{\theta}$ for $j=1,2$, and $\theta>-1$. Then for all
$A>0$

$\displaystyle\zeta(\tfrac{1}{2}+\alpha_{1}+it)\zeta(\tfrac{1}{2}+\alpha_{2}-it)=\sum_{m,n}\frac{1}{m^{1/2+\alpha_{1}}n^{1/2+\alpha_{2}}}\left(\frac{m}{n}\right)^{-it}V_{\alpha_{1},\alpha_{2}}(mn,t)$
$\displaystyle+X_{\alpha_{1},\alpha_{2},t}\sum_{m,n}\frac{1}{m^{1/2-\alpha_{2}}n^{1/2-\alpha_{1}}}\left(\frac{m}{n}\right)^{-it}V_{-\alpha_{2},-\alpha_{1}}(mn,t)+O_{A}((1+t)^{-A}).$

We can now prove the necessary twisted second moment estimate.

######  Theorem 2.2.

Let $\theta>-1$, $w$ be a smooth non-negative function supported on $[1/2,4]$,
and

$A_{\alpha_{1},\alpha_{2}}(s)=\sum_{n\leq
T^{\eta}}\frac{a_{\alpha_{1},\alpha_{2}}(n)}{n^{s}},~{}B_{\alpha_{1},\alpha_{2}}(s)=\sum_{n\leq
T^{\eta}}\frac{b_{\alpha_{1},\alpha_{2}}(n)}{n^{s}}$

be Dirichlet polynomials with $\eta<1/2$ and
$a_{\alpha_{1},\alpha_{2}}(n),b_{\alpha_{1},\alpha_{2}}(n)\ll_{\varepsilon}n^{\varepsilon}$.
Then uniformly in $\textup{Re }\alpha_{j}\ll 1/\log T$ and $|\textup{Im
}\alpha_{j}|\leq(\log T)^{\theta}$ for $j=1,2$

$\displaystyle\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+\alpha_{1}+it)\zeta($
$\displaystyle\tfrac{1}{2}+\alpha_{2}-it)A_{\alpha_{1},\alpha_{2}}(\tfrac{1}{2}+it)\overline{B_{\alpha_{1},\alpha_{2}}(\tfrac{1}{2}+it)}w(t/T)~{}dt$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\oint\displaylimits_{|z_{j}-\alpha_{j}|=3^{j}/\log T}$
$\displaystyle\frac{\zeta(1+z_{1}+z_{2})(z_{1}-z_{2})^{2}}{(z_{1}-\alpha_{1})(z_{1}-\alpha_{2})(z_{2}-\alpha_{1})(z_{2}-\alpha_{2})}F_{\alpha_{1},\alpha_{2}}(z_{1},z_{2})$
$\displaystyle\times$
$\displaystyle\int_{\mathbb{R}}Y_{\alpha_{1},\alpha_{2},t}~{}Y_{-z_{2},-z_{1},t}~{}w(t/T)~{}dt~{}dz_{1}~{}dz_{2}+O(T^{1-\delta})$

for some $\delta>0$ where

$F_{\alpha_{1},\alpha_{2}}(z_{1},z_{2})=\sum_{n,m\leq
T^{\eta}}\frac{a_{\alpha_{1},\alpha_{2}}(n)\overline{b_{\alpha_{1},\alpha_{2}}(m)}}{[n,m]}\frac{(n,m)^{z_{1}+z_{2}}}{n^{z_{2}}m^{z_{1}}}.$

###### Proof.

By Lemma 4.4, up to an arbitrary power savings in $T$ the integral of interest
equals

$\displaystyle\sum_{h,k\leq
T^{\eta}}\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{\sqrt{hk}}\sum_{m,n}\int_{\mathbb{R}}\left(\frac{hm}{kn}\right)^{-it}V_{\alpha_{1},\alpha_{2}}(mn,t)\frac{1}{m^{1/2+\alpha_{1}}n^{1/2+\alpha_{2}}}w(t/T)~{}dt$
$\displaystyle+\sum_{h,k\leq T^{\eta}}$
$\displaystyle\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{\sqrt{hk}}\sum_{m,n}\int_{\mathbb{R}}X_{\alpha_{1},\alpha_{2},t}\left(\frac{hm}{kn}\right)^{-it}V_{-\alpha_{2},-\alpha_{1}}(mn,t)\frac{1}{m^{1/2-\alpha_{2}}n^{1/2-\alpha_{1}}}w(t/T)~{}dt$

Inserting the definition of $V_{\alpha_{1},\alpha_{2}}$ we see the diagonal
terms with $hm=kn$ contribute

$\displaystyle\quad\sum_{h,k\leq
T^{\eta}}\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{2\pi
i\sqrt{hk}}\int_{\mathbb{R}}\int_{(1)}\frac{G(s)}{s}g_{\alpha_{1},\alpha_{2}}(s,t)\sum_{hm=kn}\frac{1}{m^{1/2+\alpha_{1}+s}n^{1/2+\alpha_{2}+s}}w(t/T)~{}ds~{}dt$
$\displaystyle+$ $\displaystyle\sum_{h,k\leq
T^{\eta}}\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{2\pi
i\sqrt{hk}}\int_{\mathbb{R}}X_{\alpha_{1},\alpha_{2},t}\int_{(1)}\frac{G(s)}{s}g_{-\alpha_{2},-\alpha_{1}}(s,t)\sum_{hm=kn}\frac{1}{m^{1/2-\alpha_{2}+s}n^{1/2-\alpha_{1}+s}}w(t/T)~{}ds~{}dt$

The inner Dirichlet series can be evaluated by parameterizing $m=\ell
k/(h,k)$, $n=\ell h/(h,k)$ for $\ell\geq 1$ which gives

$\sum_{hm=kn}\frac{1}{m^{1/2+\alpha_{1}+s}n^{1/2+\alpha_{2}+s}}=\frac{(h,k)^{1+\alpha_{1}+\alpha_{2}+2s}}{h^{1/2+\alpha_{2}+s}k^{1/2+\alpha_{1}+s}}\zeta(1+\alpha_{1}+\alpha_{2}+2s)$

for $\textup{Re }s=1$. Shifting the contour to $\textup{Re
}s=-1/4+\varepsilon$, we pick up a pole at $s=0$ while the zero of $G(s)$
cancels out the pole of $\zeta(1+\alpha_{1}+\alpha_{2}+2s)$. Using Stirling’s
approximation and that the shifts have small real part to bound the integral
on $\textup{Re }s=-1/4+\varepsilon$, we may simplify our expression to

$\displaystyle\sum_{h,k\leq T^{\eta}}$
$\displaystyle\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{[h,k]}\int_{\mathbb{R}}\Bigg{(}\frac{(h,k)^{\alpha_{1}+\alpha_{2}}}{h^{\alpha_{2}}k^{\alpha_{1}}}\zeta(1+\alpha_{1}+\alpha_{2})$
(5)
$\displaystyle+X_{\alpha_{1},\alpha_{2},t}\frac{(h,k)^{-\alpha_{1}-\alpha_{2}}}{h^{-\alpha_{1}}k^{-\alpha_{2}}}\zeta(1-\alpha_{1}-\alpha_{2})\Bigg{)}w(t/T)~{}dt+O\left(T^{1/2+\varepsilon}\sum_{h,k\leq
T^{\eta}}(hk)^{-1/4+\varepsilon}\right).$

The error term is $O(T^{1-\delta})$ provided $\eta<1/2$. To evaluate the main
term, we first replace $X_{\alpha_{1},\alpha_{2},t}$ by
$Y_{\alpha_{1},\alpha_{2},t}^{2}$ at a negligible cost. Because
$Y_{\alpha_{1},\alpha_{2},t}^{-1}=Y_{-\alpha_{2},-\alpha_{1},t}$, we can now
symmetrize our expression as

$\displaystyle\sum_{h,k\leq T^{\eta}}$
$\displaystyle\frac{a_{\alpha_{1},\alpha_{2}}(h)\overline{b_{\alpha_{1},\alpha_{2}}(k)}}{[h,k]}\int_{\mathbb{R}}Y_{\alpha_{1},\alpha_{2},t}\Bigg{(}Y_{-\alpha_{2},-\alpha_{1},t}\frac{(h,k)^{\alpha_{1}+\alpha_{2}}}{h^{\alpha_{2}}k^{\alpha_{1}}}\zeta(1+\alpha_{1}+\alpha_{2})$
(6)
$\displaystyle+Y_{\alpha_{1},\alpha_{2},t}\frac{(h,k)^{-\alpha_{1}-\alpha_{2}}}{h^{-\alpha_{1}}k^{-\alpha_{2}}}\zeta(1-\alpha_{1}-\alpha_{2})\Bigg{)}w(t/T)~{}dt+O\left(T^{1/2+\varepsilon}\sum_{h,k\leq
T^{\eta}}(hk)^{-1/4+\varepsilon}\right).$

Now an appeal to lemma 2.5.1 of [16] permits us to rewrite the main term as
the desired contour integral. To conclude, we must show that the off-diagonal
terms with $hm\neq kn$ give a negligible contribution provided $\eta<1/2$.
This follows by using repeated integration by parts and (4).

∎

###  2.2. Twisted fourth moment

We now cover the corresponding notation for the twisted fourth moment
estimate. Write $\sigma_{z_{1},z_{2}}(n)=\sum_{ab=n}a^{-z_{1}}b^{-z_{2}}$ for
the generalized divisor sum functions, and for the sake of concreteness let

$G(s)=e^{s^{2}}\left(1-\left(\frac{2s}{\alpha_{1}+\alpha_{3}}\right)^{2}\right)\left(1-\left(\frac{2s}{\alpha_{1}+\alpha_{4}}\right)^{2}\right)\left(1-\left(\frac{2s}{\alpha_{2}+\alpha_{3}}\right)^{2}\right)\left(1-\left(\frac{2s}{\alpha_{2}+\alpha_{4}}\right)^{2}\right).$

For simplicity of notation, denote
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$ and
$\pi\bm{\alpha}=(\alpha_{3},\alpha_{4},\alpha_{1},\alpha_{2})$. Write

$g_{\bm{\alpha}}(s,t)=g_{\alpha_{1},\alpha_{3}}(s,t)g_{\alpha_{2},\alpha_{4}}(s,t),$

$X_{\bm{\alpha},t}=X_{\alpha_{1},\alpha_{3},t}~{}X_{\alpha_{2},\alpha_{4},t},\quad
Y_{\bm{\alpha},t}=Y_{\alpha_{1},\alpha_{3},t}~{}Y_{\alpha_{2},\alpha_{4},t},$

and

$V_{\bm{\alpha}}(x,t)=\frac{1}{2\pi
i}\int_{(1)}\frac{G(s)}{s}g_{\bm{\alpha}}(s,t)x^{-s}ds.$

As before, $X_{\bm{\alpha},t}=Y_{\bm{\alpha},t}^{2}(1+O(1/t))$ and we have the
decay estimate

$t^{j}\frac{\partial^{j}}{\partial
t^{j}}V_{\bm{\alpha}}(x,t)\ll_{A,j}(1+|x|/t^{2})^{-A}.$ (7)

The appropriate approximate functional equation from theorem 5.3 of [24] takes
the following form.

######  Lemma 2.3.

Let $t\in[T/4,4T]$, $\textup{Re }\alpha_{j}\ll 1/\log T$, $|\textup{Im
}\alpha_{j}|\leq(\log T)^{\theta}$ for all $j$, and $\theta>-1$. Then for all
$A>0$

$\displaystyle\zeta(\tfrac{1}{2}+\alpha_{1}+it)\zeta(\tfrac{1}{2}+\alpha_{2}+it)\zeta(\tfrac{1}{2}+\alpha_{3}-$
$\displaystyle it)\zeta(\tfrac{1}{2}+\alpha_{4}-it)$
$\displaystyle=\sum_{m,n}\frac{\sigma_{\alpha_{1},\alpha_{2}}(m)\sigma_{\alpha_{3},\alpha_{4}}(n)}{\sqrt{mn}}\left(\frac{m}{n}\right)^{-it}$
$\displaystyle V_{\bm{\alpha}}(mn,t)$
$\displaystyle+X_{\bm{\alpha},t}\sum_{m,n}\frac{\sigma_{-\alpha_{3},-\alpha_{4}}(m)\sigma_{-\alpha_{1},-\alpha_{2}}(n)}{\sqrt{mn}}\left(\frac{m}{n}\right)^{-it}V_{-\pi\bm{\alpha}}$
$\displaystyle(mn,t)+O_{A}((1+t)^{-A}).$

Finally, we set

$B_{z_{1},z_{2},z_{3},z_{4}}(n)=\prod_{p^{m}\|n}\left(\sum_{j\geq
0}\frac{\sigma_{z_{1},z_{2}}(p^{j})\sigma_{z_{3},z_{4}}(p^{j+m})}{p^{j}}\right)\left(\sum_{j\geq
0}\frac{\sigma_{z_{1},z_{2}}(p^{j})\sigma_{z_{3},z_{4}}(p^{j})}{p^{j}}\right)^{-1},$

$A(z_{1},z_{2},z_{3},z_{4})=\frac{\zeta(1+z_{1}+z_{3})\zeta(1+z_{1}+z_{4})\zeta(1+z_{2}+z_{3})\zeta(1+z_{2}+z_{4})}{\zeta(2+z_{1}+z_{2}+z_{3}+z_{4})},$

and let

$\Delta(z_{1},z_{2},z_{3},z_{4})=\prod_{1\leq j<k\leq 4}(z_{k}-z_{j})$

denote the Vandermonde determinant.

######  Theorem 2.4.

Let $\theta>-1$, $w$ be a smooth non-negative function supported on $[1/2,4]$,
and

$A_{\bm{\alpha}}(s)=\sum_{n\leq T^{\eta}}\frac{a_{\bm{\alpha}}(n)}{n^{s}}$

be a Dirichlet polynomial with $\eta<1/11$ and
$a_{\bm{\alpha}}(n)\ll_{\varepsilon}n^{\varepsilon}$. Then uniformly for
$\textup{Re }\alpha_{j}\ll 1/\log T$ and $|\textup{Im }\alpha_{j}|\leq(\log
T)^{\theta}$

$\displaystyle\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+\alpha_{1}+it)\zeta(\tfrac{1}{2}$
$\displaystyle+\alpha_{2}+it)\zeta(\tfrac{1}{2}+\alpha_{3}-it)\zeta(\tfrac{1}{2}+\alpha_{4}-it)|A_{\bm{\alpha}}(\tfrac{1}{2}+it)|^{2}w(t/T)~{}dt$
$\displaystyle=\frac{1}{4(2\pi i)^{4}}$
$\displaystyle\oint\displaylimits_{|z_{j}-\alpha_{j}|=3^{j}/\log
T}A(\bm{z})\Delta(z_{1},z_{2},-z_{3},-z_{4})^{2}F_{\bm{\alpha}}(\bm{z})$
$\displaystyle\int_{\mathbb{R}}Y_{\bm{\alpha},t}~{}Y_{-\pi\bm{z},t}~{}w(t/T)~{}dt~{}\prod_{j\leq
4}\prod_{k\leq 4}\frac{dz_{j}}{(z_{j}-\alpha_{k})}+O(T^{1-\delta})$

for some $\delta>0$, where $\bm{z}=(z_{1},z_{2},z_{3},z_{4})$,
$-\pi\bm{z}=(-z_{3},-z_{4},-z_{1},-z_{2})$ and

$F_{\bm{\alpha}}(\bm{z})=\sum_{n,m\leq
T^{\eta}}\frac{a_{\bm{\alpha}}(n)\overline{a_{\bm{\alpha}}(m)}}{[n,m]}B_{\bm{z}}\left(\frac{n}{(n,m)}\right)B_{-\pi\bm{z}}\left(\frac{m}{(n,m)}\right).$

By using Lemma 2.3, one can prove that theorem 1.1 of [31] is also valid for
shifts with $\textup{Re }\alpha_{j}\ll 1/\log T$ and $|\textup{Im
}\alpha_{j}|\leq(\log T)^{\theta}$ provided one slightly modifies the main
term. We shall now sketch the necessary modifications.

Proceeding in the same manner of the proof of Theorem 2.2, we find that the
diagonal term $I_{D}^{(1)}(h,k)$ defined by (38) of [31] is given by

$I_{D}^{(1)}(h,k)=\frac{1}{\sqrt{hk}}\int_{\mathbb{R}}Z_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},h,k}(0)\phi(t/T)dt+O(T^{1/2+\varepsilon}(hk)^{-1/4}),$

where $Z_{\bm{\alpha},h,k}(s)$ is defined by (15) in [31]. Similarly, the term
$I_{D}^{(2)}(h,k)$ defined by (39) of [31] is given by

$I_{D}^{(2)}(h,k)=\frac{1}{\sqrt{hk}}\int_{\mathbb{R}}Y_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},t}^{2}Z_{-\alpha_{3},-\alpha_{4},-\alpha_{1},-\alpha_{2},h,k}(0)\phi(t/T)dt+O(T^{1/2+\varepsilon}(hk)^{-1/4}).$

Note we have also replaced $X_{\bm{\alpha},t}$ with $Y_{\bm{\alpha},t}^{2}$
which incurs a negligible cost. All of the $J_{\bm{\alpha}}$ terms defined by
(56) and (57) of [31] vanish since the zeros of $G$ cancel out the poles of
the integrand.

We now handle the off-diagonal contribution. Since the real parts of the
shifts are small, the error term arising from the delta method in [31] is
unchanged, but we have a slightly different main term. The key modification is
that in place of (94) of [31] one needs a more precise version of Stirling’s
approximation

$\displaystyle\frac{\Gamma(\tfrac{1}{2}-\alpha_{1}-s\pm
it)}{\Gamma(\tfrac{1}{2}+\alpha_{3}+s\pm it)}$
$\displaystyle=\exp\bigg{[}(\alpha_{1}+s+it)\log\left(1+\frac{\alpha_{1}+\alpha_{3}+2s}{\tfrac{1}{2}-\alpha_{1}-s-it}\right)+\alpha_{1}+\alpha_{3}+2s$
$\displaystyle-(\alpha_{1}+\alpha_{3}+2s$ $\displaystyle)\log(t-\textup{Im
}\alpha_{3})\mp(\alpha_{1}+\alpha_{3}+2s)\frac{\pi
i}{2}\bigg{]}\left(1+O\left(\frac{(1+|s|^{2})(\log
T)^{2\theta}}{t}\right)\right),$

which holds uniformly in $|\alpha_{1}|,|\alpha_{3}|\leq(\log T)^{\theta}<T/10$
and $t\in[T/4,4T]$. Notice also that the hypotheses of lemma 2 of [8] are also
satisfied in this range, so we may write

$\displaystyle
X_{\alpha_{1},\alpha_{3},t}=\lambda(\tfrac{1}{2}+\alpha_{1}+it)\lambda(\tfrac{1}{2}+\alpha_{3}-it)=$
$\displaystyle\left(1+O\left(\frac{(\log T)^{2\theta}}{t}\right)\right)$
$\displaystyle\times\exp\bigg{(}-(\alpha_{1}+\alpha_{3})\log\left(\frac{t-\textup{Im
}\alpha_{3}}{2\pi}\right)+(\alpha_{1}+$ $\displaystyle
it)\log\left(1+\frac{\alpha_{1}+\alpha_{3}}{\tfrac{1}{2}-\alpha_{1}-it}\right)+\alpha_{1}+\alpha_{3}\bigg{)}.$

One may then continue following the argument of section 6 of [31] with
$X_{\alpha_{1},\alpha_{3},t}$ in place of $(t/2\pi)^{-\alpha_{1}-\alpha_{3}}$.
The expression for $P_{\bm{\alpha}}^{(0)}$ in proposition 6.8 of [31] will now
be replaced by

$P_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{(0)}=\frac{1}{\sqrt{hk}}\int_{\mathbb{R}}Y_{\alpha_{1},\alpha_{3},t}^{2}~{}Z_{-\alpha_{3},\alpha_{2},-\alpha_{1},\alpha_{4},h,k}(0)\phi(t/T)dt,$

where again we have replaced $X_{\alpha_{1},\alpha_{3},t}$ with
$Y_{\alpha_{1},\alpha_{3},t}^{2}$ at a negligible cost.

Upon adding up all the diagonal and off-diagonal terms, we now conclude the
proof by symmetrizing the integrand as we did for Theorem 2.2. Simply note
that we can divide out a factor of $Y_{\bm{\alpha}}$ from the integrand by
using that $Y_{\gamma,\delta,t}^{-1}=Y_{-\delta,-\gamma,t}$ and

$Y_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}^{-1}=Y_{-\alpha_{3},-\alpha_{1},t}~{}Y_{-\alpha_{4},-\alpha_{2},t}=Y_{-\alpha_{4},-\alpha_{1},t}~{}Y_{-\alpha_{3},-\alpha_{2},t}.$

This puts the integrand in a form where lemma 2.5.1 of [16] is applicable, and
we may conclude by summing over $h$ and $k$.

##  3\. Upper Bounds

To begin our proof, we will first note a simple bound we will use when the
shifts $h_{1}$ and $h_{2}$ in (3) are close to one another.

######  Proposition 3.1.

For $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$ and $0\leq\beta\leq 1$

$\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}\ll(\log
T)^{4\beta^{2}}.$

###### Proof.

This is an immediate consequence of the Cauchy Schwarz inequality and the
bound

$\frac{1}{T}\int_{T/4}^{4T}|\zeta(\tfrac{1}{2}+it)|^{4\beta}\ll(\log
T)^{4\beta^{2}},$

for $0\leq\beta\leq 1$ which follows from theorem 1 of [29].

∎

To handle larger shifts, stronger techniques are needed since
$\zeta(\tfrac{1}{2}+it+ih_{1})$ and $\zeta(\tfrac{1}{2}+it+ih_{2})$ decouple.
We will use twisted moments to efficiently estimate the correlation.
Throughout this section we will select the following parameters: Denote by
$\log_{j}$ the $j$-fold iterated logarithm, and we will take $\ell$ to be the
largest integer so that $\log_{\ell}T\geq 10^{4}$. Let $T_{0}=e^{2}$ and for
$1\leq j\leq\ell$ take

$T_{j}=\exp\left(\frac{\log T}{(\log_{j+1}T)^{2}}\right)$

and $K_{j}=250P_{j}$ for $1\leq j\leq\ell$.

The first step will be to bound the integrand in (3) by products of integral
powers of zeta and Dirichlet polynomials following [29].

######  Proposition 3.2.

For $0\leq\beta\leq 1$ and $s,w\in\mathbb{C}$

$\displaystyle|\zeta(s)|^{2\beta}|\zeta(w)|^{2\beta}$
$\displaystyle\ll|\zeta(s)|^{2}|\zeta(w)|^{2}\prod_{1\leq
j\leq\ell}|\mathcal{N}_{j}(s;\beta-1)|^{2}|\mathcal{N}_{j}(w;\beta-1)|^{2}$
$\displaystyle+\prod_{1\leq
j\leq\ell}|\mathcal{N}_{j}(s;\beta)|^{2}|\mathcal{N}_{j}(w;\beta)$
$\displaystyle|^{2}+\sum_{1\leq
v\leq\ell}\bigg{(}|\zeta(s)|^{2}|\zeta(w)|^{2}\prod_{1\leq
j<v}|\mathcal{N}_{j}(s;\beta-1)|^{2}|\mathcal{N}_{j}(w;\beta-1)|^{2}$
$\displaystyle+\prod_{1\leq j<v}$
$\displaystyle|\mathcal{N}_{j}(s;\beta)|^{2}|\mathcal{N}_{j}(w;\beta)|^{2}\bigg{)}\left(\Bigg{|}\frac{\mathcal{P}_{v}(s)}{50P_{v}}\Bigg{|}^{2\lceil
50P_{v}\rceil}+\Bigg{|}\frac{\mathcal{P}_{v}(w)}{50P_{v}}\Bigg{|}^{2\lceil
50P_{v}\rceil}\right).$

###### Proof.

Take $1\leq v\leq\ell$ to be the smallest index such that either
$|\mathcal{P}_{v}(s)|>50P_{v}$ or $|\mathcal{P}_{v}(w)|>50P_{v}$, and set
$v=\ell+1$ if no such index exists. Applying Young’s inequality $ab\leq
a^{p}/p+b^{q}/q$ for conjugate exponents $p=1/\beta$, $q=1/(1-\beta)$ gives

$|\zeta(s)|^{2\beta}|\zeta(w)|^{2\beta}\prod_{1\leq
j<v}e^{2(1-\beta)(\textup{Re }\mathcal{P}_{j}(s)+\textup{Re
}\mathcal{P}_{j}(w))}\ll|\zeta(s)|^{2}|\zeta(w)|^{2}+\prod_{1\leq
j<v}e^{2(\textup{Re }\mathcal{P}_{j}(s)+\textup{Re }\mathcal{P}_{j}(w))},$

hence

$|\zeta(s)|^{2\beta}|\zeta(w)|^{2\beta}\ll|\zeta(s)|^{2}|\zeta(w)|^{2}\prod_{1\leq
j<v}e^{2(\beta-1)(\textup{Re }\mathcal{P}_{j}(s)+\textup{Re
}\mathcal{P}_{j}(w))}+\prod_{1\leq j<v}e^{2\beta(\textup{Re
}\mathcal{P}_{j}(s)+\textup{Re }\mathcal{P}_{j}(w))}.$

The next step is to expand the exponentials into Dirichlet polynomials. Since
$|\mathcal{P}_{j}(s)|\leq 50P_{j}$ and $|\mathcal{P}_{j}(w)|\leq 50P_{j}$ for
$j<v$, lemma 1 of [29] furnishes the bound

$\displaystyle|\zeta(s)|^{2\beta}|\zeta(w)|^{2\beta}$
$\displaystyle\ll|\zeta(s)|^{2}|\zeta(w)|^{2}\prod_{1\leq
j<v}|\mathcal{N}_{j}(s;\beta-1)|^{2}|\mathcal{N}_{j}(w;\beta-1)|^{2}(1-e^{-P_{j}})^{-2}$
$\displaystyle+\prod_{1\leq
j<v}|\mathcal{N}_{j}(s;\beta-1)|^{2}|\mathcal{N}_{j}(w;\beta-1)|^{2}(1-e^{-P_{j}})^{-2}.$

To conclude, note that when $1\leq v\leq\ell$ we may multiply the right-hand
side by

$\Bigg{|}\frac{\mathcal{P}_{v}(s)}{50P_{v}}\Bigg{|}^{2\lceil
50P_{v}\rceil}+\Bigg{|}\frac{\mathcal{P}_{v}(w)}{50P_{v}}\Bigg{|}^{2\lceil
50P_{v}\rceil}\geq 1.$

Since $\prod_{1\leq j\leq\ell}(1-e^{-P_{j}})^{-2}\leq 4$, the claim follows by
summing over $1\leq v\leq\ell+1$.

∎

Therefore proving Theorem 1.1 is now reduced to a handful of moment
calculations. To simplify the notation, we will write

$\mathcal{N}_{h_{1},h_{2},j}(s;\beta):=\mathcal{N}_{j}(s+ih_{1};\beta)\mathcal{N}_{j}(s+ih_{2};\beta).$

######  Proposition 3.3.

If $\beta\leq 1$, $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$, and
$|h_{1}-h_{2}|\geq 200/\log T$, then for large $T$

$\displaystyle\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+i$ $\displaystyle
h_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2}\prod_{1\leq
j\leq\ell}|\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta-1)|^{2}dt$ (8)
$\displaystyle\ll(\log
T)^{2\beta^{2}}|\zeta(1+i(h_{1}-h_{2}))|^{2}|\zeta(1+1/\log
T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}-2}$

and for $1\leq v\leq\ell$, $k\in\\{1,2\\}$ and $0\leq r\leq\lceil
50P_{v}\rceil$

$\displaystyle\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})$
$\displaystyle\zeta(\tfrac{1}{2}+it+ih_{2})|^{2}\prod_{1\leq
j<v}|\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta-1)|^{2}|\mathcal{P}_{v}(\tfrac{1}{2}+i(t+h_{k}))|^{2r}dt$
$\displaystyle\ll(\log T)^{2}$ $\displaystyle(\log
T_{v-1})^{2\beta^{2}-2}|\zeta(1+i(h_{1}-h_{2}))|^{2}$
$\displaystyle\times|\zeta(1+1/\log
T_{v-1}+i(h_{1}-h_{2}))|^{2\beta^{2}-2}\left(18^{r}r!P_{v}^{r}\exp(P_{v})\right).$

######  Proposition 3.4.

If $\beta\leq 1$, $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$, and
$|h_{1}-h_{2}|\geq 200/\log T$, then for large $T$

$\frac{1}{T}\int_{T}^{2T}\prod_{1\leq
j\leq\ell}|\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta)|^{2}dt\ll(\log
T)^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}$ (9)

and for $1\leq v\leq\ell$, $k\in\\{1,2\\}$ and $0\leq r\leq\lceil
50P_{v}\rceil$

$\displaystyle\frac{1}{T}$ $\displaystyle\int_{T}^{2T}\prod_{1\leq
j<v}|\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta)|^{2}|\mathcal{P}_{v}(\tfrac{1}{2}+i(t+h_{k}))|^{2r}dt$
$\displaystyle\ll(\log T_{v-1})^{2\beta^{2}}|\zeta(1+1/\log
T_{v-1}+i(h_{1}-h_{2}))|^{2\beta^{2}}\left(r!P_{v}^{r}\right).$

Before embarking on the proofs of these propositions, we first show how to
deduce Theorem 1.1 from them.

###### Proof of Theorem 1.1.

Split the range of integration in (3) depending on whether $|h_{1}-h_{2}|\leq
200/\log T$ or $|h_{1}-h_{2}|>200/\log T$. By Proposition 3.1, the
contribution of the region where $|h_{1}-h_{2}|\leq 200/\log T$ to the
integral (3) is $\ll(\log T)^{4\beta^{2}+\theta-1}$.

To handle the region where $|h_{1}-h_{2}|>200/\log T$, Propositions 3.2, 3.3,
3.4, and an application of Fubini’s theorem shows

$\displaystyle\frac{1}{T}\iint\displaylimits_{\begin{subarray}{c}|h_{1}|,|h_{2}|\leq(\log
T)^{\theta}\\\ |h_{1}-h_{2}|>200/\log
T\end{subarray}}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})$
$\displaystyle\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}~{}dt~{}dh_{1}~{}dh_{2}\ll(\log
T)^{2\beta^{2}+\theta}(I_{\ell}^{(1)}+I_{\ell}^{(2)})$ $\displaystyle+(\log
T)^{\theta}\sum_{1\leq v\leq\ell}(\log T_{v-1})^{2\beta^{2}}$
$\displaystyle\Bigg{(}\left(\frac{\log T}{\log
T_{v-1}}\right)^{2}\frac{18^{\lceil 50P_{v}\rceil}\lceil
50P_{v}\rceil!P_{v}^{\lceil 50P_{v}\rceil}\exp(P_{v})}{(50P_{v})^{2\lceil
50P_{v}\rceil}}I_{v-1}^{(1)}$ $\displaystyle+\frac{\lceil
50P_{v}\rceil!P_{v}^{\lceil 50P_{v}\rceil}}{(50P_{v})^{2\lceil
50P_{v}\rceil}}I_{v-1}^{(2)},\Bigg{)}$

where we have set

$I_{v}^{(1)}=\int_{200/\log T}^{2(\log T)^{\theta}}|\zeta(1+1/\log
T_{v}+ih)|^{2\beta^{2}-2}|\zeta(1+ih)|^{2}~{}dh,$ $I_{v}^{(2)}=\int_{200/\log
T}^{2(\log T)^{\theta}}|\zeta(1+1/\log T_{v}+ih)|^{2\beta^{2}}~{}dh.$

First suppose $-1<\theta\leq 0$. Then we may use the Laurent expansion of
$\zeta$ near 1 to estimate

$I_{v}^{(1)}\ll\int_{200/\log T}^{2(\log T)^{\theta}}(1/(\log
T_{v})^{2}+h^{2})^{1-\beta^{2}}h^{-2}dh.$

Because $0\leq\beta\leq 1$ the integrand is $\ll(\log
T_{v})^{2\beta^{2}-2}h^{-2}+h^{-2\beta^{2}}$, so

$\displaystyle I_{v}^{(1)}$ $\displaystyle\ll\log T(\log
T_{v})^{2\beta^{2}-2}+\int_{200/\log T}^{(\log T)^{\theta}}h^{-2\beta^{2}}dh$
$\displaystyle\ll\log T(\log
T_{v})^{2\beta^{2}-2}+\begin{cases}\frac{1}{2\beta^{2}-1}(\log
T)^{2\beta^{2}-1}&1/\sqrt{2}<\beta\leq 1,\\\ \frac{1}{1-2\beta^{2}}(\log
T)^{\theta-2\beta^{2}\theta}&0\leq\beta<1/\sqrt{2},\\\ (1+\theta)\log\log
T&\beta=1/\sqrt{2}.\end{cases}$

For $I_{v}^{(2)}$ we have the simpler bound

$\displaystyle I_{v}^{(2)}$ $\displaystyle\ll\int_{200/\log T}^{2(\log
T)^{\theta}}(1/(\log T_{v})^{2}+h^{2})^{-\beta^{2}}~{}dh\ll\int_{200/\log
T}^{2(\log T)^{\theta}}h^{-2\beta^{2}}~{}dh$
$\displaystyle\ll\begin{cases}\frac{1}{2\beta^{2}-1}(\log
T)^{2\beta^{2}-1}&1/\sqrt{2}<\beta\leq 1,\\\ \frac{1}{1-2\beta^{2}}(\log
T)^{\theta-2\beta^{2}\theta}&0\leq\beta<1/\sqrt{2},\\\ (1+\theta)\log\log
T&\beta=1/\sqrt{2}.\end{cases}$

We can now conclude the proof in the case of $-1<\theta\leq 0$ by summing over
$v$, noting that the $\log T(\log T_{v})^{2\beta-2}$ terms appearing in the
upper bound for $I_{v}^{(1)}$ contribute

$\displaystyle\ll(\log T)^{\theta}$ $\displaystyle\sum_{1\leq v\leq\ell}(\log
T_{v-1})^{4\beta^{2}-4}(\log T)^{3}\exp(-50P_{v})$ $\displaystyle\ll(\log
T)^{4\beta^{2}+\theta-1}\sum_{1\leq v\leq\ell}\left(\frac{\log T_{v-1}}{\log
T}\right)^{4\beta^{2}-1}\left(\frac{\log T}{\log
T_{v-1}}\right)^{3}\exp(-50P_{v})$ $\displaystyle\ll(\log
T)^{4\beta^{2}+\theta-1}\sum_{1\leq v\leq\ell}\left(\frac{\log T}{\log
T_{v-1}}\right)^{4}\exp(-50P_{v})\ll(\log T)^{4\beta^{2}+\theta-1},$

which is admissible.

Now suppose $\theta>0$. Break the range of integration of $I_{v}^{(1)}$ and
$I_{v}^{(2)}$ into the regions $1/\log T\leq h\leq 1$ and $1\leq h\leq(\log
T)^{\theta}$. For $1/\log T\leq h\leq 1$, the argument for $-1<\theta\leq 0$
still goes through, and the contribution to $I_{v}^{(1)}+I_{v}^{(2)}$ is of
order

$\ll\log T(\log T_{v})^{2\beta^{2}-2}+\begin{cases}\frac{1}{2\beta^{2}-1}(\log
T)^{2\beta^{2}-1}&1/\sqrt{2}<\beta\leq 1,\\\
\frac{1}{1-2\beta^{2}}&0\leq\beta<1/\sqrt{2},\\\ \log\log
T&\beta=1/\sqrt{2}.\end{cases}$

Then using moment bounds for the zeta function to the right of the one line
[44, Theorems 7.1 and 7.2], the contribution of $1\leq h\leq(\log T)^{\theta}$
to $I_{v}^{(1)}+I_{v}^{(2)}$ is of order $\ll(\log T)^{\theta}$. The implied
constant here is absolute because $0\leq\beta\leq 1$. Overall, the conclusion
is

$I_{v}^{(1)}+I_{v}^{(2)}\ll(\log T)^{\theta}+\log T(\log
T_{v})^{2\beta^{2}-2}+\begin{cases}\frac{1}{2\beta^{2}-1}(\log
T)^{2\beta^{2}-1}&1/\sqrt{2}<\beta\leq 1,\\\
\frac{1}{1-2\beta^{2}}&0\leq\beta<1/\sqrt{2},\\\ \log\log
T&\beta=1/\sqrt{2}.\end{cases}$

The case $\theta>0$ of Theorem 1.2 now follows by summing over $1\leq
v\leq\ell$.

∎

### Proof of Propositions 3.3 and 3.4

We will apply on Theorem 2.4 with
$\bm{\alpha}=(ih_{1},ih_{2},-ih_{1},-ih_{2})$ and $w(t)$ a smooth majorant of
$1_{t\in[1,2]}$ to the Dirichlet polynomials

$\prod_{1\leq j\leq\ell}\mathcal{N}_{h_{1},h_{2},j}(s;\beta-1)$ (10)

and

$\prod_{1\leq j\leq
v}\mathcal{N}_{h_{1},h_{2},j}(s;\beta-1)\mathcal{P}_{v}(\tfrac{1}{2}+i(t+h_{k}))^{r}$
(11)

for $k\in\\{1,2\\}$. First note that since $|h_{1}-h_{2}|\geq 200/\log T$, on
the contours of the $z_{j}$ we have the bounds

$A(\bm{z})\ll(\log
T)^{2}|\zeta(1+i(h_{1}-h_{2}))|^{2},\quad\Delta(z_{1},z_{2},-z_{3},-z_{4})^{2}\ll(\log
T)^{-4}|h_{1}-h_{2}|^{8},$

and because $\textup{Re }z\ll 1/\log T$ it readily follows that

$\int_{\mathbb{R}}Y_{\bm{\alpha},t}~{}Y_{-\pi{\bm{z}},t}~{}w(t/T)~{}dt\ll T.$

Therefore, by multiplicativity we are reduced to bounding the size of the sums

$F_{h_{1},h_{2},j}(\bm{z})=\sum_{n,m\leq
T^{\eta}}\frac{a_{h_{1},h_{2},j}(n)\overline{a_{h_{1},h_{2},j}(m)}}{[n,m]}B_{\bm{z}}\left(\frac{n}{(n,m)}\right)B_{\pi\bm{z}}\left(\frac{m}{(n,m)}\right)$

and

$F_{v,r,h_{k}}(\bm{z})=\sum_{n,m\leq
T^{\eta}}\frac{b_{v,r,h_{k}}(n)\overline{b_{v,r,h_{k}}(m)}}{[n,m]}B_{\bm{z}}\left(\frac{n}{(n,m)}\right)B_{\pi\bm{z}}\left(\frac{m}{(n,m)}\right),$

uniformly for $|z_{j}-\alpha_{j}|\leq 200/\log T$, where
$a_{h_{1},h_{2},j}(n)$ are the coefficients of the Dirichlet polynomials
$\mathcal{N}_{h_{1},h_{2},j}(s)$, and $b_{v,r,h_{k}}(n)$ are the coefficients
of $\mathcal{P}_{v}(s+ih_{k})^{2r}$.

First we estimate $F_{h_{1},h_{2},j}(\bm{z})$. Note we can write

$\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta-1)=\sum_{\begin{subarray}{c}p\mid
n\Rightarrow p\in(T_{j-1},T_{j}]\\\ \Omega(n)\leq
K_{j}\end{subarray}}\frac{1}{n^{1/2+it}}\sum_{cd=n}\frac{(\beta-1)^{\Omega(c)+\Omega(d)}g(c)g(d)}{c^{ih_{1}}d^{ih_{2}}},$

so
$a_{h_{1},h_{2},j}(p)=(\beta-1)(p^{-ih_{1}}+p^{-ih_{2}})=(\beta-1)\sigma_{ih_{1},ih_{2}}(p)$.
Since $\beta\leq 1$ it also follows that $|a_{h_{1},h_{2},j}(n)|\leq d(n)$,
where $d$ is the divisor function. We will require the following estimates for
$B_{\bm{z}}(n)$.

######  Lemma 3.5.

For $m\geq 1$

$\displaystyle
B_{\bm{\alpha}}(p^{m})=\frac{\sigma_{\alpha_{3},\alpha_{4}}(p^{m})-\sigma_{\alpha_{3},\alpha_{4}}(p^{m-1})p^{-1-\alpha_{3}-\alpha_{4}}(p^{-\alpha_{1}}+p^{-\alpha_{2}})+\sigma_{\alpha_{3},\alpha_{4}}(p^{m-2})p^{-2-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}}}{1-p^{-2-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}}},$

where by convention we set $\sigma_{\alpha_{3},\alpha_{4}}(p^{-1})=0$.
Furthermore, for integers $n$ composed of primes at most $T^{10^{-8}}$ and
$\textup{Re }\alpha_{j}\ll 1/\log T$ for each $j$

$|B_{\bm{\alpha}}(n)|\ll d_{3}(n),$

where $d_{3}$ is the ternary divisor function.

###### Proof.

The first formula follows from lemma 6.9 of [31] and that

$\sigma_{z_{1},z_{2}}(p^{m})=\frac{p^{-z_{3}(m+1)}-p^{-z_{4}(m+1)}}{p^{-z_{3}}-p^{-z_{4}}}.$

To prove the second bound, first note by assumption on the size of the shifts
and the size of the primes $p$

$|\sigma_{\alpha_{3},\alpha_{4}}(p^{m})|\leq\sum_{ab=p^{m}}a^{\textup{Re
}\alpha_{3}}b^{\textup{Re }\alpha_{4}}\leq(1+m)\left(1+O\left(\frac{\log
p}{\log T}\right)\right).$

Whence

$|B_{\bm{\alpha}}(p^{m})|\leq d(p^{m})\left(1+O\left(\frac{\log p}{\log
T}+\frac{1}{p}\right)\right).$

It now follows by assumption on $n$ that

$|B_{\bm{\alpha}}(n)|\ll d(n)\prod_{p^{m}\|n}\left(1+O\left(\frac{m\log
p}{\log T}\right)\right)\left(1+O\left(\frac{1}{p}\right)\right)\ll d_{3}(n)$

where we have used $\prod_{p^{m}\|n}\left(1+O\left(\frac{m\log p}{\log
T}\right)\right)\ll\exp(\log n/\log T)$ and $\prod_{p\mid
n}\left(1+O\left(\frac{1}{p}\right)\right)\ll(3/2)^{\omega(n)}$, where
$\omega(n)$ is the number of distinct prime divisors of $n$.

∎

The following lemma is almost identical to the proof of Lemma 24 of [2]. The
proof is a straightforward application of Cauchy’s theorem. The only necessary
modification is to use Lemma 3.5 to bound $B_{\bm{z}}(p)$ because here $z$
need not have small imaginary part.

######  Lemma 3.6.

Let $m\geq 1$ be an integer and $\bm{z}=(z_{1},z_{2},z_{3},z_{4})$,
$\bm{w}=(w_{1},w_{2},w_{3},w_{4})$ vectors such that $|w_{j}-z_{j}|\leq
200/\log T$ for all $j$. Then

$|B_{\bm{z}}(p)-B_{\bm{w}}(p)|\ll\frac{\log p}{\log T}.$

We will now bound $F_{h_{1},h_{2},j}(\bm{z})$ by a product over primes. To
accomplish this, we will drop the terms with $\Omega(m),\Omega(n)\leq K_{j}$
and use Rankin’s trick to show the error incurred is negligible. First note
that $\exp(\Omega(n)+\Omega(n)-K_{j})\geq 1$ when either $\Omega(n)$ or
$\Omega(m)$ exceed $K_{j}$. Now using Lemma 3.5 and that $|\beta-1|\leq 1$, we
see that adding in the terms of $F_{h_{1},h_{2},j}(\bm{z})$ with
$\Omega(m)>K_{j}$ or $\Omega(n)>K_{j}$ contributes an error of at most

$\displaystyle\ll e^{-K_{j}}\sum_{p|m,n\Rightarrow
p\in(T_{j-1},T_{j}]}\frac{(|\beta-1|e)^{\Omega(n)+\Omega(m)}}{[n,m]}d_{3}(n)d_{3}(m)$
$\displaystyle\ll e^{-K_{j}}\prod_{T_{j-1}<p\leq
T_{j}}\left(1+\frac{6e+9e^{2}}{p}+O\left(\frac{1}{p^{2}}\right)\right)\ll
e^{-70P_{j}}$

Therefore upon dropping the assumption that $\Omega(m),\Omega(n)\leq K_{j}$
and using Lemmas 3.5, 3.6, and that $|a_{h_{1},h_{2}}(n)|\leq d(n)$, we may
bound $F_{h_{1},h_{2},j}(\bm{z})$ by

$\displaystyle\prod_{T_{j-1}<p\leq
T_{j}}\bigg{(}1+\frac{a_{h_{1},h_{2},j}(p)B_{ih_{1},ih_{2},-ih_{1},-ih_{2}}(p)+a_{-h_{1},-h_{2},j}(p)B_{-ih_{1},-ih_{2},ih_{1},ih_{2}}(p)+|a_{h_{1},h_{2}}(p)|^{2}}{p}$
$\displaystyle+O\left(\frac{\log p}{p\log
T}+\frac{1}{p^{2}}\right)\bigg{)}+O(e^{-70P_{j}})\qquad\qquad\qquad$

By Lemma 3.5, the numerator of the second term in parentheses equals

$\frac{(\beta-1)(\beta-1+p+\beta p)}{1+p}(2+2\cos((h_{1}-h_{2})\log
p))=(2\beta^{2}-2)(1+\cos((h_{1}-h_{2})\log p))+O\left(\frac{1}{p}\right).$

Therefore, we must control the products

$\displaystyle\prod_{T_{j-1}<p\leq
T_{j}}\bigg{(}1+\frac{(2\beta^{2}-2)(1+\cos((h_{1}-h_{2})\log
p))}{p}+O\left(\frac{\log p}{p\log T}+\frac{1}{p^{2}}\right)\bigg{)}$

To this end, we will need the following special case of lemma 3.2 of [35].

######  Lemma 3.7.

Given $h\in\mathbb{R}$ and $X\geq 2$

$\displaystyle\sum_{p\leq X}$ $\displaystyle\frac{\cos(h\log
p)}{p}=\log|\zeta(1+1/\log X+ih)|+O(1).$

######  Remark.

While similar results have appeared in previous literature, this formulation
is due to Granville and Soundararajan [22].

###### Proof.

Because the Euler product of $\zeta(s)$ is convergent for $\textup{Re }s>1$ we
may write

$\displaystyle\log|\zeta(1+1/\log X+ih)|$ $\displaystyle=\textup{Re
}\left(\sum_{p}\frac{1}{p^{1+1/\log X+ih}}+\sum_{p}\sum_{m\geq
2}\frac{1}{mp^{m(1+1/\log X+ih)}}\right)$
$\displaystyle=\sum_{p}\frac{\cos(h\log p)}{p^{1+1/\log X}}+O(1).$

The primes larger than $X$ contribute at most

$\sum_{p>X}\frac{1}{p^{1+1/\log X}}\ll 1.$

For $p\leq X$ the mean value theorem gives $p^{1/\log X}=1+O(\log p/\log X)$,
so the claim follows from Merten’s first estimate $\sum_{p\leq X}\frac{\log
p}{p}=\log X+O(1).$ ∎

Therefore uniformly for $|z_{j}-\alpha_{j}|\leq 200/\log T$, the product
$\prod_{1\leq j<v}F_{h_{1},h_{2},j}(\bm{z})$ is of order at most

$\displaystyle\prod_{1\leq j<v}\Bigg{(}\prod_{T_{j-1}<p\leq
T_{j}}\bigg{(}1+\frac{(2\beta^{2}-2)(1+\cos((h_{1}-h_{2})\log p))}{p}$
$\displaystyle+O\left(\frac{\log p}{p\log
T}+\frac{1}{p^{2}}\right)\bigg{)}+O(e^{-70P_{j}})\Bigg{)}$
$\displaystyle\ll(\log T_{v-1})^{2\beta^{2}-2}|\zeta(1+1/\log T_{v-1}$
$\displaystyle+i(h_{1}-h_{2}))|^{2\beta^{2}-2}.$

Since $\log T_{\ell}\asymp\log T$, this proves the first bound in Proposition
3.3. All that remains is to estimate the $F_{v,r,h_{k}}(\bm{z})$. Note that
$|d_{3}(n)|\ll 3^{\Omega(n)}$ and we can trivially estimate $|n^{-ih_{k}}|\leq
1$, so

$|F_{v,r,h_{k}}(\bm{z})|\leq 9^{r}\sum_{\begin{subarray}{c}p|m,n\Rightarrow
p\in(T_{v-1},T_{v}]\\\
\Omega(n)=\Omega(m)=r\end{subarray}}\frac{r!^{2}g(n)g(m)}{[n,m]}.$

In the proof of proposition 3 of [29], it was shown that the right-hand side
of is at most $18^{r}r!P_{v}^{r}\exp(P_{v})$, which concludes the proof of the
second bound of Proposition 3.3.

To handle the first bound in Proposition 3.4, we will use the mean value
theorem for Dirichlet polynomials [24, Theorem 9.1] in place of Theorem 2.4.
We will abuse of notation and also denote the coefficients of
$N_{h_{1},h_{2},j}(s;\beta)$ by $a_{h_{1},h_{2},j}(n)$ now with $\beta$ in
place of $\beta-1$. Therefore

$\displaystyle\frac{1}{T}\int_{T}^{2T}\prod_{1\leq
j\leq\ell}|\mathcal{N}_{h_{1},h_{2},j}(\tfrac{1}{2}+it;\beta)|^{2}dt\ll\prod_{1\leq
j\leq\ell}\sum_{\begin{subarray}{c}p\mid n\Rightarrow p\in(T_{j-1},T_{j}]\\\
\Omega_{j}(n)\leq K_{j}\forall
j\end{subarray}}\frac{|a_{h_{1},h_{2},j}(n)|^{2}}{n}.$

Now the same reasoning as the proof of Proposition 3.3 shows we may bound this
expression by

$\displaystyle\ll\prod_{1\leq j\leq\ell}\Bigg{(}\prod_{T_{j-1}<p\leq
T_{j}}\bigg{(}1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log p))}{p}+$
$\displaystyle O\left(\frac{\log p}{p\log
T}+\frac{1}{p^{2}}\right)\bigg{)}+O(e^{-70P_{j}})\Bigg{)}$
$\displaystyle\ll(\log T_{\ell})^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}$
$\displaystyle-h_{2}))|^{2\beta^{2}}.$

To prove the second bound of Proposition 3.4, note that the mean value theorem
for Dirichlet polynomials and the same reasoning as the proof of Proposition
3.3 now gives a bound of

$\displaystyle\ll\prod_{1\leq j<v}\Bigg{(}\prod_{T_{j-1}<p\leq
T_{j}}\bigg{(}1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log p))}{p}+$
$\displaystyle O\left(\frac{\log p}{p\log
T}+\frac{1}{p^{2}}\right)\bigg{)}+O(e^{-70P_{j}})\Bigg{)}$
$\displaystyle\times\sum_{\begin{subarray}{c}p|n\Rightarrow
p\in(T_{v-1},T_{v}]\\\ \Omega(n)=r\end{subarray}}\frac{(r!g(n))^{2}}{n}$
$\displaystyle\ll(\log T_{v-1})^{2\beta^{2}}|\zeta(1+1/\log T_{v-1}+i(h_{1}$
$\displaystyle-h_{2}))|^{2\beta^{2}}(r!P_{v}^{r}).$

Here we have used that the sum over $n$ above is bounded by $r!P_{v}^{r}$,
which is a porism of proposition 2 of [29]. This concludes the proofs of
Propositions 3.3 and 3.4, so Theorem 1.1 follows. ∎

##  4\. Lower Bounds

Theorem 1.2 will follow from the following proposition

######  Proposition 4.1.

Uniformly in $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$ and $200/\log
T\geq|h_{1}-h_{2}|$

$\frac{1}{T}\int_{T}^{2T}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}~{}dt\gg_{\beta}(\log
T)^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}.$

Indeed Theorem 1.2 follows by integrating the bound given by Proposition 4.1
over the range $200/\log T\leq|h_{1}-h_{2}|\leq 2(\log T)^{\theta}$ using the
Laurent expansion of $\zeta$ and standard moment estimates for $\zeta$ to the
right of the one line. In view of Proposition 3.1, it is perhaps not so
surprising that the contribution of small $|h_{1}-h_{2}|$ is not necessary to
give a lower bound of the correct order. We will first prove Proposition 4.1
in the case $\beta\leq 1$.

###  4.1. Proof of Proposition 4.1 for $\beta\leq 1$

Throughout this subsection, we will let $T_{0}=e^{2}$, let $\ell$ be the
largest integer such that $\log_{\ell}T\geq 10^{4},$ and for $1\leq j\leq\ell$
set

$T_{j}=\exp\left(\frac{\beta\log T}{(\log_{j}T)^{2}}\right)$

and $K_{j}=250P_{j}$. With this choice of parameters, $\mathcal{N}(s;\alpha)$
is a Dirichlet polynomial of length at most $T^{\beta/18}$. Inspired by the
method of Heap and Soundararajan [30], our key inequality is the following
consequence of Hölder’s inequality:

$\displaystyle\Bigg{|}\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it$
$\displaystyle+ih_{2})\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)$
(12) $\displaystyle\quad\times\mathcal{N}(\tfrac{1}{2}-it-
ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}-it-ih_{2};\beta)w(t/T)~{}dt\Bigg{|}$
$\displaystyle\leq\bigg{(}\int_{\mathbb{R}}$
$\displaystyle|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}w(t/T)~{}dt\bigg{)}^{\tfrac{1}{2}}$
$\displaystyle\times\bigg{(}\int_{\mathbb{R}}|\zeta(\tfrac{1}{2}+it+ih_{1})$
$\displaystyle\zeta(\tfrac{1}{2}+it+ih_{2})|^{2}$
$\displaystyle\times|\mathcal{N}(\tfrac{1}{2}+it$
$\displaystyle+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)|^{2}w(t/T)~{}dt\bigg{)}^{\tfrac{1-\beta}{2}}$
$\displaystyle\times\bigg{(}\int_{\mathbb{R}}|\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)$
$\displaystyle\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)|^{2}$
$\displaystyle\times|\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta)|^{2/\beta}w(t/T)~{}dt\bigg{)}^{\tfrac{\beta}{2}},$

where $0\leq w(t)\leq 1$ is a smooth function supported on the interval
$[1.1,1.9]$ with $w(t)=1$ for $t\in[1.2,1.8]$, say. Note that Proposition 3.3
gives the upper bound

$\displaystyle\int_{\mathbb{R}}|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2}$
$\displaystyle|\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)|^{2}w(t/T)~{}dt$
$\displaystyle\ll T(\log T)^{2\beta^{2}}|\zeta(1+1/\log
T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}$

so we are left with two moment computations.

######  Remark.

It is also natural to try to prove Theorem 1.2 by computing the twisted moment

$\displaystyle\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}-it-
ih_{2})$
$\displaystyle\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta)$
$\displaystyle\times\mathcal{N}(\tfrac{1}{2}-it-
ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}-it-ih_{2};\beta-1)w(t/T)~{}dt.$

While this would allow us to use Theorem 2.2, it seems more difficult to
evaluate this integral asymptotically.

Before carrying out these computations, it will be convenient to introduce
some notation. For $1\leq j\leq\ell$, we set

$A_{h_{1},h_{2},j}(\tfrac{1}{2}+it):=\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{2};\beta-1),$
$B_{h_{1},h_{2},j}(\tfrac{1}{2}+it):=\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{1};\beta)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{2};\beta).$

These are Dirichlet polynomials supported on integers $n$ such that
$\Omega(n)\leq K_{j}$ and $n$ is composed of primes $p\in(T_{j-1},T_{j}]$ with
coefficients

$a_{h_{1},h_{2},j}(n):=\sum_{cd=n}\frac{(\beta-1)^{\Omega(c)+\Omega(d)}g(c)g(d)}{c^{ih_{1}}d^{ih_{2}}},$
$b_{h_{1},h_{2},j}(n):=\sum_{cd=n}\frac{\beta^{\Omega(c)+\Omega(d)}g(c)g(d)}{c^{ih_{1}}d^{ih_{2}}}.$

Additionally, for $a=1,2$ define

$M_{a}(t)=|\mathcal{N}(\tfrac{1}{2}+it+ih_{a};\beta-1)||\mathcal{N}(\tfrac{1}{2}+it+ih_{a};\beta)|^{1/\beta}.$

Proposition 4.1 will follow from the following two propositions.

######  Proposition 4.2.

Uniformly in $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$ and $|h_{1}-h_{2}|\geq
200/\log T$

$\displaystyle\Bigg{|}\frac{1}{T}\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}-it-
ih_{2})\prod_{1\leq j\leq\ell}$ $\displaystyle
A_{h_{1},h_{2},j}(\tfrac{1}{2}+it)\overline{B_{h_{1},h_{2},j}(\tfrac{1}{2}+it)}w(t/T)~{}dt~{}\Bigg{|}$
$\displaystyle\gg_{\beta}(\log T)^{2\beta^{2}}|\zeta(1+1/\log
T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}.$

######  Proposition 4.3.

Uniformly in $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$ and $|h_{1}-h_{2}|\geq
200/\log T$

$\frac{1}{T}\int_{\mathbb{R}}M_{1}M_{2}(t)^{2}w(t/T)dt\ll_{\beta}(\log
T)^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}.$

#### Proof of Proposition 4.2

The task at hand is to evaluate the integral

$\displaystyle\mathcal{I}\beta,h_{1},h_{2})=\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it$
$\displaystyle+ih_{2})\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)$
$\displaystyle\times\mathcal{N}(\tfrac{1}{2}-it-
ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}-it-ih_{2};\beta)w(t/T)~{}dt.$

We will accomplish this with an appropriate approximate functional equation.
Set

$\widetilde{g}_{\alpha_{1},\alpha_{2}}(s,t)=\pi^{-s}\frac{\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{1}+s+it}{2}\right)\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{2}+s+it}{2}\right)}{\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{1}+it}{2}\right)\Gamma\left(\frac{\tfrac{1}{2}+\alpha_{2}+it}{2}\right)},$
$\widetilde{X}_{\alpha_{1},\alpha_{2},t}=\lambda(\tfrac{1}{2}+\alpha_{1}+it)\lambda(\tfrac{1}{2}+\alpha_{2}+it),$

and

$\widetilde{V}_{\alpha_{1},\alpha_{2}}(x,t)=\frac{1}{2\pi
i}\int_{(1)}\frac{e^{s^{2}}}{s}\widetilde{g}_{\alpha_{1},\alpha_{2}}(s,t)x^{-s}~{}ds,$

which satisfies the decay estimate

$t^{j}\frac{\partial^{j}}{\partial
t^{j}}\widetilde{V}_{\alpha_{1},\alpha_{2}}(x,t)\ll_{A,j}(1+|x|/t)^{-A}.$

######  Lemma 4.4 ([24] Theorem 5.3).

Let $t\in[T,2T]$, $\textup{Re }\alpha_{j}\ll 1/\log T$ and $|\textup{Im
}\alpha_{j}|\leq(\log T)^{\theta}$ for $j=1,2$ and $\theta>-1$. Then for all
$A>0$

$\displaystyle\zeta(\tfrac{1}{2}+\alpha_{1}+it)\zeta(\tfrac{1}{2}+\alpha_{2}+it)=\sum_{m,n}\frac{1}{m^{1/2+\alpha_{1}}n^{1/2+\alpha_{2}}}\left(mn\right)^{-it}\widetilde{V}_{\alpha_{1},\alpha_{2}}(mn,t)$
$\displaystyle+\widetilde{X}_{\alpha_{1},\alpha_{2},t}\sum_{m,n}\frac{1}{m^{1/2-\alpha_{2}}n^{1/2-\alpha_{1}}}\left(mn\right)^{it}\widetilde{V}_{-\alpha_{2},-\alpha_{1}}(mn,t)+O_{A}((1+t)^{-A}).$

Therefore, up to a negligible error, that $I(\beta,h_{1},h_{2})$ is given by

$\displaystyle\prod_{j\leq\ell}\sum_{h,k,m,n}\frac{a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(k)}}{m^{h_{1}}n^{h_{2}}\sqrt{hkmn}}\int_{\mathbb{R}}\left(\frac{hmn}{k}\right)^{-it}\widetilde{V}_{h_{1},h_{2}}(mn,t)w(t/T)~{}dt$
(13) $\displaystyle+\prod_{j\leq\ell}$
$\displaystyle\sum_{h,k,m,n}\frac{a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(k)}}{m^{-h_{2}}n^{-h_{1}}\sqrt{hkmn}}\int_{\mathbb{R}}\widetilde{X}_{h_{1},h_{2},t}\left(\frac{h}{kmn}\right)^{-it}\widetilde{V}_{-h_{2},-h_{1}}(mn,t)w(t/T)~{}dt.$

Using the decay of $\widetilde{V}_{\alpha_{1},\alpha_{2}}$, we can discard the
contribution of terms with $hmn\neq k$ and $h\neq kmn$ in the first and second
lines of (13) respectively. Next we can parameterize the diagonal sums in the
first and second lines by $\sum_{h|k,mn=k/h}$ and $\sum_{k|h,mn=h/k}$
respectively. Then after using the defition of $\widetilde{V}_{\alpha,\beta}$,
shifting the contour, and recognizing the sum over $m$ and $n$ as a familiar
Dirichlet convolution we find

$\displaystyle\mathcal{I}(\beta,$ $\displaystyle
h_{1},h_{2})=\prod_{j\leq\ell}\sum_{h|k}\frac{a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(k)}}{k}\sigma_{h_{1},h_{2}}(k/h)\int_{\mathbb{R}}w(t/T)~{}dt$
(14)
$\displaystyle+\prod_{j\leq\ell}\sum_{k|h}\frac{a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(k)}}{h}\sigma_{-h_{2},-h_{1}}(h/k)\int_{\mathbb{R}}\widetilde{X}_{h_{1},h_{2},t}~{}w(t/T)~{}dt+O(T^{1-\delta}).$

for some $\delta>0$.

Due to the oscillating term $\widetilde{X}_{h_{1},h_{2},t}$ the second line in
(14) will not contribute to the leading order, and we will control this term
first. Because $w(t/T)$ is supported on $[T,2T]$ and that
$|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$, Stirling’s approximation gives
$\widetilde{X}_{h_{1},h_{2},t}=e^{-if(t)}(1+O(1/T))$ where

$f(t)=(h_{1}+t)\log\left(\frac{t+h_{1}}{2\pi}\right)+(h_{2}+t)\log\left(\frac{t+h_{2}}{2\pi}\right)-2t-h_{1}-h_{2}-\frac{\pi}{2}.$

For $t\in[T,2T]$ note that $f^{\prime}(t)\gg\log t$ and that
$f^{\prime\prime}(t)\ll 1/t$. Therefore, an integration by parts à la van der
Corput gives the bound

$\int_{\mathbb{R}}\widetilde{X}_{h_{1},h_{2},t}~{}w(t/T)~{}dt\ll 1/\log T.$

Therefore because $\prod_{1\leq
j\leq\ell}A_{h_{1},h_{2},j}(s)\overline{B_{h_{1},h_{2},j}(s)}$ is a Dirichlet
polynomial of length $\leq T^{\beta/9}$, we may absorb the second line of (14)
into the error term $O(T^{1-\delta})$.

It remains to analyze the sums

$S_{j}:=\sum_{h|k}\frac{a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(k)}}{k}\sigma_{h_{1},h_{2}}(k/h)=\sum_{a,h}\frac{\sigma_{h_{1},h_{2}}(a)a_{h_{1},h_{2},j}(h)\overline{b_{h_{1},h_{2},j}(ah)}}{ah}$

for $j\leq\ell$. Here we are summing over all integers $a,h$ composed of
primes in $(T_{j-1},T_{j}]$ such that $\Omega(a)+\Omega(h)\leq K_{j}$. If we
discard the latter condition on $a$ and $h$ via Rankin’s trick, we incur an
error at most

$\displaystyle e^{-K_{j}}$
$\displaystyle\prod_{p\in(T_{j-1},T_{j}]}\sum_{r,s\geq
0}\frac{|\sigma_{h_{1},h_{2}}(p^{r})a_{h_{1},h_{2},j}(p^{s})\overline{b_{h_{1},h_{2},j}(p^{r+s})}|}{p^{r+s}}e^{\Omega(r)+\Omega(s)}$
$\displaystyle\ll
e^{-K_{j}}\prod_{p\in(T_{j-1},T_{j}]}\left(1+\frac{2e}{p}+O\left(\frac{1}{p^{2}}\right)\right)\ll
e^{-100P_{j}},$

whence

$S_{j}=\prod_{p\in(T_{j-1},T_{j}]}\sum_{r,s\geq
0}\frac{\sigma_{h_{1},h_{2}}(p^{r})a_{h_{1},h_{2},j}(p^{s})\overline{b_{h_{1},h_{2},j}(p^{r+s})}}{p^{r+s}}+O(e^{-100P_{j}}).$

Now since $a_{h_{1},h_{2},j}(p)=(\beta-1)\sigma_{h_{1},h_{2}}(p)$ and
$b_{h_{1},h_{2},j}(p)=\beta\sigma_{h_{1},h_{2}}(p)$ it follows that

$S_{j}=\prod_{p\in(T_{j-1},T_{j}]}\left(1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log
p))}{p}+O\left(\frac{1}{p^{2}}\right)\right)+O(e^{-100P_{j}}),$

so up to a power savings $\mathcal{I}(\beta,h_{1},h_{2})$ is equal to

$T\prod_{j\leq\ell}\left(\prod_{p\in(T_{j-1},T_{j}]}\left(1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log
p))}{p}+O\left(\frac{1}{p^{2}}\right)\right)+O(e^{-100P_{j}})\right)\|w\|_{1}.$

Using Lemma 3.7 and that $\|w\|_{1}>0.8$ we conclude

$|\mathcal{I}(\beta,h_{1},h_{2})|\gg_{\beta}T(\log
T)^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}-h_{2}))|,$

and Proposition 4.2 follows. ∎

#### Proof of Proposition 4.3

Since we only need an upper bound, we will bound the weight $w(t/T)$ by the
characteristic function of $[T,2T]$. Therefore, we must simply evaluate the
mean value of the Dirichlet polynomial $M_{1}M_{2}(t)$ on $[T,2T]$. We will
decompose each

$M_{a}(t)=\prod_{j\leq\ell}M_{a,j}(t)$

where

$M_{a,j}(t):=|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta-1)||\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta)|^{1/\beta}.$

Following [30] we will now bound $|M_{1,j}M_{2,j}|^{2}$ by products of genuine
Dirichlet polynomials.

######  Lemma 4.5.

For $j\leq\ell$

$\displaystyle|M_{1,j}(t)M_{2,j}(t)|^{2}\leq|\mathcal{N}_{j}(\tfrac{1}{2}+it$
$\displaystyle+ih_{1};\beta)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$
$\displaystyle+O\big{(}2^{2/\beta}Q_{1,j}(t)|\mathcal{N}_{j}(\tfrac{1}{2}$
$\displaystyle+it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$
$\displaystyle+2^{2/\beta}Q_{2,j}(t)|\mathcal{N}_{j}(\tfrac{1}{2}+$
$\displaystyle it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$
$\displaystyle+2^{4/\beta}$ $\displaystyle Q_{1,j}(t)Q_{2,j}(t)\big{)},$

where for $a=1,2$

$Q_{a,j}(t):=\left(\frac{12|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{K_{j}}\right)^{2K_{j}}\sum_{r=0}^{K_{j}/\beta}\left(\frac{2e|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{r+1}\right)^{2r}$

###### Proof.

The proof is nearly identical to the proof of lemma 1 of [30]. If
$|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|\leq K_{j}/10$, then we may write

$\displaystyle|M_{a,j}(t)|^{2}$ $\displaystyle=\exp\left(2\beta\textup{Re
}\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})+\right)(1+O(e^{-K_{j}/10}))^{2}$
$\displaystyle=|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta)|^{2}(1+O(e^{-K_{j}/10})).$

In the case $|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|>K_{j}/10$ for some $a$
then the proof of lemma 1 of [30] provides the bounds

$|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta-1)|\leq\left(\frac{12|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{K_{j}}\right)^{2K_{j}}$

and

$|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta)|^{1/\beta}\ll
2^{2/\beta}\sum_{r=0}^{K_{j}/\beta}\left(\frac{2e|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{r+1}\right)^{2r}.$

The claim now follows by summing over all four cases where
$|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{1})|$ and
$|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{2})|$ are either smaller or larger than
$K_{j}/10$. ∎

######  Lemma 4.6.

For $a=1,2$

$\frac{1}{T}\int_{T}^{2T}Q_{a,j}(t)dt\ll
e^{-K_{j}},\quad\frac{1}{T}\int_{T}^{2T}Q_{a,j}(t)^{2}dt\ll e^{-K_{j}}.$

###### Proof.

To handle the first estimate, note for $0\leq r\leq K_{j}/\beta$

$\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})^{K_{j}+r}=\sum_{\begin{subarray}{c}\Omega(n)=K_{j}+r\\\
p|n\Rightarrow
p\in(T_{j-1},T_{j}]\end{subarray}}\frac{(K_{j}+r)!g(n)}{n^{1/2+it+ih_{a}}}.$

This is a Dirichlet polynomial of length at most $T_{j}^{K_{j}(1+1/\beta)}$
whose coefficients have magnitude $(K_{j}+r)!g(n)/n^{1/2}$, so the claim
follows by the same argument used in lemma 2 of [30]. The only difference is
that here $K_{j}=250P_{j}$.

To handle the second estimate an application of the Cauchy Schwarz inequality
yields

$Q_{a,j}(t)^{2}\leq\left(\frac{12|P_{j}(\tfrac{1}{2}+it+ih_{a})|}{K_{j}}\right)^{4K_{j}}\cdot\frac{K_{j}}{\beta}\sum_{r=0}^{K_{j}/\beta}\left(\frac{2e|P_{j}(\tfrac{1}{2}+it+ih_{a})|}{r+1}\right)^{4r}.$

Next note that

$\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})^{2K_{j}+2r}=\sum_{\begin{subarray}{c}\Omega(n)=2K_{j}+2r\\\
p|n\Rightarrow
p\in(T_{j-1},T_{j}]\end{subarray}}\frac{(2K_{j}+2r)!g(n)}{n^{1/2+it+ih_{a}}},$

which is also a short Dirichlet polynomial. Now the mean value theorem for
Dirichlet polynomials gives

$\frac{1}{T}\int_{T}^{2T}|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|^{4K_{j}+4r}dt\ll\sum_{\begin{subarray}{c}\Omega(n)=2K_{j}+2r\\\
p|n\Rightarrow
p\in(T_{j-1},T_{j}]\end{subarray}}\frac{(2K_{j}+2r)!^{2}g(n)^{2}}{n}\ll(2K_{j}+2r)!P_{j}^{2K_{j}+2r},$

where we have used that $g(n)\leq 1$ and the definition of $P_{j}$. Whence

$\frac{1}{T}\int_{T}^{2T}Q_{a,j}(t)^{2}dt\ll\frac{K_{j}}{\beta}\left(\frac{12}{K_{j}}\right)^{4K_{j}}\sum_{r=0}^{K_{j}/\beta}\left(\frac{2e}{r+1}\right)^{4r}(2K_{j}+2r)!P_{j}^{2K_{j}+2r}.$

The summand is maximized near $r$ satisfying $r^{2}=4P_{j}(2K_{j}+2r).$ Since
$K_{j}=250P_{j}$ any such $r$ necessarily lies in
$[2\sqrt{P_{j}K_{j}},2.1\sqrt{P_{j}K_{j}}]$, and we conclude in the same
manner as lemma 2 of [30].

∎

To conclude, we will use the following splitting lemma, which appears in
equation (16) of [30]

######  Lemma 4.7.

Suppose for $1\leq j\leq\ell$ we have $j$ disjoint intervals $I_{j}$ and
Dirichlet polynomials $A_{j}(s)=\sum_{n}a_{j}(n)n^{-s}$ such that $a_{j}(n)$
vanishes unless $n$ is composed of primes in $I_{j}$. Then if
$\prod_{j\leq\ell}A_{j}(s)$ is a Dirichlet polynomial of length $\leq N$, then

$\displaystyle\frac{1}{T}\int_{T}^{2T}\prod_{j\leq\ell}|A_{j}(\tfrac{1}{2}+it)|^{2}dt=(1+O(N/T))\prod_{j\leq\ell}\left(\frac{1}{T}\int_{T}^{2T}|A_{j}(\tfrac{1}{2}+it)|^{2}dt\right)$

In [30] equation (16), there is an additional factor of $\log N$ in the error
term which arises because [30] uses a simpler treatment of the mean value
theorem for Dirichlet polynomials. This however can be easily removed by using
a version of the mean value theorem for Dirichlet polynomials with a slightly
stronger error term, i.e. theorem 9.1 of [24]. In fact a version of this
splitting lemma has also appeared in [2, lemma 14].

To conclude the proof, note the mean value theorem for Dirichlet polynomials
gives

$\displaystyle\frac{1}{T}\int_{T}^{2T}$
$\displaystyle|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{1};\beta)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{2};\beta)|^{2}dt$
$\displaystyle=(1+O(T^{-8/9}))\sum_{\begin{subarray}{c}p|n\Rightarrow
p\in(T_{j-1},T_{j}]\\\ \Omega(n)\leq
K_{j}\end{subarray}}\frac{|b_{h_{1},h_{2},j}(n)|^{2}}{n}$
$\displaystyle\leq(1+O(T^{-8/9}))\prod_{p\in(T_{j-1},T_{j}]}\left(1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log
p))}{p}+O\left(\frac{1}{p^{2}}\right)\right).$ (15)

Similar reasoning gives that for $a=1,2$

$\displaystyle\frac{1}{T}\int_{T}^{2T}|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta)|^{2}dt$
$\displaystyle\leq(1+O(T^{-8/9}))\prod_{p\in(T_{j-1},T_{j}]}\left(1+\frac{\beta^{2}}{p}+O\left(\frac{1}{p^{2}}\right)\right)$
$\displaystyle\leq$ $\displaystyle(1+O(T^{-8/9}))\left(\frac{\log T_{j}}{\log
T_{j-1}}\right)^{\beta^{2}}.$ (16)

Combining these calculations with lemma 4.5, 4.6, and 4.7 and bounding
integrals of products using the Cauchy Schwarz inequality, we find

$\displaystyle\frac{1}{T}\int_{\mathbb{R}}M_{1}M_{2}(t)^{2}w(t/T)dt$
$\displaystyle\ll_{\beta}\prod_{1\leq j\leq\ell}\Bigg{(}\prod_{T_{j-1}<p\leq
T_{j}}\Bigg{(}1+\frac{2\beta^{2}(1+\cos((h_{1}-h_{2})\log p))}{p}$
$\displaystyle+O\left(\frac{1}{p^{2}}\right)\Bigg{)}+O(e^{-K_{j}/10})\Bigg{)}$
$\displaystyle\ll_{\beta}(\log T)^{2\beta^{2}}|\zeta(1+1/\log
T_{\ell}+i(h_{1}$ $\displaystyle-h_{2}))|^{2\beta^{2}}.$

This completes the proof of Proposition 4.3.

###  4.2. Proof of Proposition 4.1 for $\beta\geq 1$

The case of $\beta\geq 1$ a bit simpler than the case $\beta\leq 1$. Now let
$T_{0}=\beta^{4}e^{2}$, let $\ell$ be the largest integer such that
$\log_{\ell}T\geq 10^{4},$ for $1\leq j\leq\ell$ set

$T_{j}=\exp\left(\frac{\log T}{\beta^{2}(\log_{j}T)^{2}}\right),$

and take $K_{j}=250\beta^{2}P_{j}$. With this choice of parameters,
$\mathcal{N}(s;\alpha)$ is a Dirichlet polynomial of length at most
$T^{1/18}$. Now because $\beta\geq 1$, Hölder’s inequality gives

$\displaystyle\Bigg{|}\int_{\mathbb{R}}\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it$
$\displaystyle+ih_{2})\mathcal{N}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)$
(17) $\displaystyle\quad\times\mathcal{N}(\tfrac{1}{2}-it-
ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}-it-ih_{2};\beta)w(t/T)~{}dt\Bigg{|}$
$\displaystyle\leq\bigg{(}\int_{\mathbb{R}}$
$\displaystyle|\zeta(\tfrac{1}{2}+it+ih_{1})\zeta(\tfrac{1}{2}+it+ih_{2})|^{2\beta}w(t/T)~{}dt\bigg{)}^{\tfrac{1}{2\beta}}$
$\displaystyle\times\bigg{(}\int_{\mathbb{R}}|\mathcal{N}(\tfrac{1}{2}+it+ih_{1};$
$\displaystyle\beta-1)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta-1)$
$\displaystyle\times\mathcal{N}(\tfrac{1}{2}$
$\displaystyle+it+ih_{1};\beta)\mathcal{N}(\tfrac{1}{2}+it+ih_{2};\beta)|^{\frac{2\beta}{2\beta-1}}w(t/T)~{}dt\bigg{)}^{1-\tfrac{1}{2\beta}}.$

Therefore, if we denote

$N_{a,j}(t):=|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{1};\beta-1)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{1};\beta)|^{\tfrac{2\beta}{2\beta-1}}$

all we must show is

######  Proposition 4.8.

For $\beta\geq 1$, $|h_{1}|,|h_{2}|\leq(\log T)^{\theta}$ and
$|h_{1}-h_{2}|\geq 200/\log T$

$\displaystyle\frac{1}{T}\int_{\mathbb{R}}\prod_{j\leq\ell}$ $\displaystyle
N_{1,j}(t)N_{2,j}(t)w(t/T)~{}dt$ $\displaystyle\ll_{\beta}(\log
T)^{2\beta^{2}}|\zeta(1+1/\log T_{\ell}+i(h_{1}-h_{2}))|^{2\beta^{2}}.$

This is a lot like Proposition 4.3, but the proof is even simpler. The analog
of Lemma 4.5 is

######  Lemma 4.9.

For $j\leq\ell$

$\displaystyle N_{1},j(t)N_{2,j}(t)\leq|\mathcal{N}_{j}(\tfrac{1}{2}+it$
$\displaystyle+ih_{1};\beta)\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$
$\displaystyle+O\big{(}R_{1,j}(t)|\mathcal{N}_{j}(\tfrac{1}{2}$
$\displaystyle+it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$
$\displaystyle+R_{2,j}(t)|\mathcal{N}_{j}(\tfrac{1}{2}+$ $\displaystyle
it+ih_{2};\beta)|^{2}(1+O(e^{-K_{j}/10}))$ $\displaystyle+$ $\displaystyle
R_{1,j}(t)R_{2,j}(t)\big{)},$

where for $a=1,2$

$R_{a,j}(t):=\left(\frac{12\beta|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{K_{j}}\right)^{2K_{j}}.$

###### Proof.

Following the proof of Lemma 4.5, if
$|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|\leq K_{j}/10$, then we may write

$\displaystyle
N_{a,j}(t)=|\mathcal{N}_{j}(\tfrac{1}{2}+it+ih_{a};\beta)|^{2}(1+O(e^{-K_{j}/10})).$

In the case $|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|>K_{j}/10$ for some $a$
then simply note for $\alpha=\beta-1$ or $\alpha=\beta$

$\displaystyle|\mathcal{N}_{j}(\tfrac{1}{2}$
$\displaystyle+it+ih_{a};\alpha)|^{\tfrac{2\beta}{2\beta-1}}\leq\left(\sum_{r=0}^{K_{j}}\frac{\beta^{r}|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|^{r}}{r!}\right)^{\tfrac{2\beta}{2\beta-1}}$
$\displaystyle\leq\left(\sum_{r=0}^{K_{j}}\frac{\beta^{r}|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|^{r}}{r!}\right)^{2}\leq\left(\frac{12\beta|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|}{K_{j}}\right)^{2K_{j}},$

where we have used that $\beta\geq 1$ and that
$|\mathcal{P}_{j}(\tfrac{1}{2}+it+ih_{a})|$ is large. The claim now follows. ∎

The analog of Lemma 4.5 is

######  Lemma 4.10.

For $a=1,2$

$\frac{1}{T}\int_{T}^{2T}R_{a,j}(t)dt\ll
e^{-K_{j}},\quad\frac{1}{T}\int_{T}^{2T}R_{a,j}(t)^{2}dt\ll e^{-K_{j}}.$

The proof is a simpler version of Lemma 4.5, and is omitted. Now we may
conclude the proof of Proposition 4.8 when $\beta\geq 1$ by combining Lemma
4.9, Lemma 4.10 and Lemma 4.7 with the mean value theorem for Dirichlet
polynomials as we did in the $\beta\leq 1$ case. This concludes the proof of
4.8, hence the proof of Theorem 1.2.

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