capbtabboxtable[][]

# Patterns of Primes in Joint Sato–Tate Distributions

A. Anas Chentouf  Massachusetts Institute of Technology, Cambridge, MA 02139,
USA<EMAIL_ADDRESS>,  Catherine H. Cossaboom  University of Virginia,
Charlottesville, VA 22903, USA<EMAIL_ADDRESS>,  Samuel E. Goldberg
University of Virginia, Charlottesville, VA 22903, USA<EMAIL_ADDRESS>and  Jack B. Miller  Yale University, New Haven, CT 06511, USA
<EMAIL_ADDRESS>
###### Abstract.

For $j=1,2$, let $f_{j}(z)=\sum_{n=1}^{\infty}a_{j}(n)e^{2\pi inz}$ be a
holomorphic, non-CM cuspidal newform of even weight $k_{j}\geq 2$ with trivial
nebentypus. For each prime $p$, let $\theta_{j}(p)\in[0,\pi]$ be the angle
such that $a_{j}(p)=2p^{(k-1)/2}\cos\theta_{j}(p)$. The now-proven Sato–Tate
conjecture states that the angles $(\theta_{j}(p))$ equidistribute with
respect to the measure
$d\mu_{\mathrm{S}T}=\frac{2}{\pi}\sin^{2}\theta\,d\theta$. We show that, if
$f_{1}$ is not a character twist of $f_{2}$, then for subintervals
$I_{1},I_{2}\subseteq[0,\pi]$, there exist infinitely many bounded gaps
between the primes $p$ such that $\theta_{1}(p)\in I_{1}$ and
$\theta_{2}(p)\in I_{2}$. We also prove a common generalization of the bounded
gaps with the Green–Tao theorem.

##  1\. Introduction

The twin prime conjecture states that there are infinitely many pairs of
consecutive primes $p_{n}$ and $p_{n+1}$ which differ exactly by $2$. Although
the twin prime conjecture remains unproven, partial progress has been made
over the past several decades. A consequence of the prime number theorem is
that $p_{n}\sim n\log p_{n}$, so the average gap $p_{n+1}-p_{n}$ is
asymptotically $\log p_{n}$, i.e., unbounded. In 2005, Goldston, Pintz, and
Yıldırım [16] developed a sieve method that detects tuples of primes in short
intervals, demonstrating that

(1.1) $\liminf_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=0.$

In particular, the result (1.1) says that for all $\varepsilon>0$, there exist
infinitely many $n$ such that $p_{n+1}-p_{n}\leq\varepsilon\log p_{n}$. That
is, there exist infinitely many consecutive primes whose gap is an arbitrarily
small multiple of the mean gap. In 2013, Zhang [49] significantly improved
upon the GPY argument, showing that

$\liminf_{n\to\infty}(p_{n+1}-p_{n})\leq 70{,}000{,}000.$

Later that year, Maynard brought the value of $70{,}000{,}000$ down to $600$
[29]. He also proved that if $m\in\mathbb{N}$, then

$\liminf_{n\to\infty}(p_{n+m}-p_{n})\ll m^{3}e^{4m}.$

(Tao developed the underlying ideas independently, but arrived at slightly
different conclusions.) Later that year, the Polymath Research Project [36]
synthesized the arguments of Zhang, Maynard, and Tao, proving that

$\liminf\limits_{n\to\infty}(p_{n+1}-p_{n})\leq 246.$

More generally, the existence of infinitely many bounded gaps has been
established within special sets of primes of number-theoretic interest. See
[2, 43, 45], for example.

In this paper, we are interested in bounded gaps between primes related to the
Sato–Tate conjecture. Let $f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)e^{2\pi
inz}\in S_{k}^{\rm new}(\Gamma_{0}(N))$ be a holomorphic cusp form without
complex multiplication (CM) and with trivial nebentypus, level $N$, and even
integral weight $k\geq 2$. We normalize $f$ so that $\lambda_{f}(1)=1$, and
the trivial nebentypus condition then implies that
$\lambda_{f}(n)\in\mathbb{R}$ for all $n$. If $f$ is also an eigenform for all
of the Hecke operators and all of the Atkin–Lehner involutions $|_{k}W(N)$ and
$|_{k}W(Q_{p})$ for each prime $p\mid N$, then $f$ is a newform (see [33,
Section 2.5]). For any prime $p$, Deligne’s proof of the Weil conjectures
implies that $|\lambda_{f}(p)|\leq 2p^{(k-1)/2}$. Thus, there exists a unique
angle $\theta_{f}(p)\in[0,\pi]$ such that
$\lambda_{f}(p)={2p^{(k-1)/2}}\cos{\theta_{f}(p)}$.

It is natural to ask how the angles $\theta_{f}(p)$ distribute as the prime
$p$ varies. Given an interval $I\subseteq[0,\pi]$, we consider the set of
primes

$\mathcal{P}_{f,I}:=\\{p:p\nmid N,\theta_{f}(p)\in I\\}.$

One may ask how many primes $p\leq x$ belong to $\mathcal{P}_{f,I}$.
Specifically, we are interested in the prime counting function

$\pi_{f,I}(x):=\\#(\mathcal{P}_{f,I}\cap[1,x]).$

The Sato–Tate conjecture, in the form proposed by Serre [40] which holds for
all non-CM newforms as above, is equivalent to the statement that for the
measure $d\mu_{\rm ST}:=\frac{2}{\pi}\sin^{2}\theta\,d\theta$,

(1.2) $\lim_{x\to\infty}\frac{\pi_{f,I}(x)}{\pi(x)}=\mu_{\rm ST}(I).$

This was proven by Barnet-Lamb, Geraghty, Harris, and Taylor [3]. See the
works of Thorner [44] and Hoey, Iskander, Jin, and Trejos-Suárez [21] for
effective bounds on the rate of convergence in (1.2).

Given a pair of distinct non-CM holomorphic newforms
$f_{j}(z)=\sum_{n=1}^{\infty}\lambda_{j}(n)e^{2\pi inz}\in
S_{k_{j}}^{\textrm{new}}(\Gamma_{0}(N_{j}))$, $j=1,2$, with
$\lambda_{1}(1)=\lambda_{2}(1)=1$, it is natural to ask what is the joint
distribution of the angles $\theta_{1}(p)$ and $\theta_{2}(p)$. For a pair of
subintervals $I_{1},I_{2}\subseteq[0,\pi]$, we consider the set of primes

(1.3) $\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}:=\\{p:p\nmid N_{1}N_{2},\
\theta_{1}(p)\in I_{1},\ \theta_{2}(p)\in I_{2}\\}.$

Next, we define the joint prime counting function

$\pi_{f_{1},f_{2},I_{1},I_{2}}(x):=\\#(\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}\cap[1,x]).$

A natural question to ask, first posed by Mazur and Katz, is whether

(1.4)
$\lim_{x\to\infty}\frac{\pi_{f_{1},f_{2},I_{1},I_{2}}(x)}{\pi(x)}=\mu_{\rm
ST}(I_{1})\mu_{\rm ST}(I_{2}).$

That is, when do the angles $\theta_{1}(p)$ and $\theta_{2}(p)$ behave
asymptotically like independent random variables?

We impose a technical condition on the relationship between $f_{1}$ and
$f_{2}$ in order to obtain the asymptotic (1.4). We say that $f_{1}$ and
$f_{2}$ are _twist-inequivalent_ , denoted by $f_{1}\not\sim f_{2}$, if for
every primitive Dirichlet character $\chi$ one has that $f_{1}\neq
f_{2}\otimes\chi$. Building on the work of Harris [20], Wong [48] proved that
(1.4) holds when $f_{1}\not\sim f_{2}$. See Thorner [44] for an effective
bound on the rate of convergence. In this paper, we establish the infinitude
of bounded gaps within the set of primes
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$ defined in (1.3).

######  Theorem 1.1.

For $j=1,2$, let $f_{j}(z)\in S_{k_{j}}^{\rm new}(\Gamma_{0}(N_{j}))$ be non-
CM holomorphic cuspidal newforms with trivial nebentypus that are unrelated by
twist. If $I_{1},I_{2}\subseteq[0,\pi]$ are subintervals and $P_{n}$ denotes
the $n$-th prime in $\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$, then

$\liminf_{n\to\infty}(P_{n+m}-P_{n})\ll\frac{N_{1}N_{2}m}{\varphi(N_{1}N_{2})(\mu_{\rm
ST}(I_{1})\mu_{\rm
ST}(I_{2}))^{8/3}}\exp\Big{(}\frac{4581N_{1}N_{2}m}{\varphi(N_{1}N_{2})(\mu_{\rm
ST}(I_{1})\mu_{\rm ST}(I_{2}))^{8/3}}\Big{)}.$

The implied constant is absolute and effective.

In 2004, Green and Tao [17] proved that the primes form infinitely many
arithmetic progressions of arbitrary length. This was achieved by extending
Szemerédi’s theorem to the primes via a transference principle. Subsequently,
many simplifications and improvements have been made to this argument,
including a criterion of Pintz [34].

Since $\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$ has positive density in the
primes, the Green–Tao theorem implies the existence of infinitely many
arithmetic progressions of arbitrarily length in
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$. In this paper, we prove a common
generalization of infinitely many bounded gaps with the Green–Tao theorem.
That is, there exist infinitely many arithmetic progressions of arbitrary
length in $\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$ which occur simultaneously
as the shifts of an admissible set. We postpone the full statement of this
result until Theorem 4.8 in Section 4.

## Acknowledgements

The authors were participants in the 2023 UVA REU in Number Theory. They are
grateful for support from Jane Street Capital, the National Science Foundation
(DMS-2002265 and DMS- 2147273), the National Security Agency
(H98230-23-1-0016), the Massachusetts Institute of Technology, and the
Templeton World Charity Foundation. The authors would like to thank Jesse
Thorner for advising this project and for many helpful conversations, as well
as Ken Ono for his valuable comments.

##  2\. Conventions and Notation

Let $\mathbb{N}$ denote the set of positive integers. Let $\mathbb{P}$ denote
the set of prime numbers $\\{2,3,5,\ldots\\}$. Throughout the paper, $p$
always denotes a prime. Given a set $S$, we let $\mathbbm{1}_{S}$ denote the
indicator function of $S$.

We let $a\mid b$ denote the proposition that $a$ divides $b$, and $a\nmid b$
to be its negation. Although the notation $[a,b]$ is also used for intervals
in the paper, we also use $[a,b]$ to denote the least common multiple of
$a,b\in\mathbb{N}$. Similarly, we will also use $(a,b)$ to denote the greatest
common divisor. In both cases, it will be clear from context which of the two
meanings is intended. We say that a finite subset
$\mathcal{H}\subseteq\mathbb{Z}$ is admissible if for every prime $p$, the
reduction of $\mathcal{H}\ (\mathrm{mod}\ p)$ is a proper subset of
$\mathbb{Z}/p\mathbb{Z}$. The diameter of an admissible set $\mathcal{H}$ is
${\rm diam}(\mathcal{H}):=\max_{1\leq i,j\leq k}|h_{j}-h_{j}|$. We assume
without loss of generality in our results that $0\in\mathcal{H}$.

Given a natural number $n\in\mathbb{N}$, it admits a prime factorization
$n=p_{1}^{\alpha_{1}}\cdots p_{r}^{\alpha_{r}}$ where $p_{1}<\cdots<p_{r}$. We
then define the following arithmetic functions:

  * •

$P^{-}(n):=p_{1}$, is the smallest prime divisor of $n$;

  * •

$\operatorname{rad}(n):=p_{1}\cdots p_{r}$, the product of the distinct primes
dividing $n$;

  * •

$\Lambda(n)$ is the von Mangoldt function, defined as $\Lambda(n)=\log{p}$ if
$n=p^{\ell}$ for some prime $p$ and $\ell\in\mathbb{N}$, and $\Lambda(n)=0$
otherwise;

  * •

$d_{k}(n):=\prod_{j=1}^{r}\binom{k+\alpha_{j}-1}{k-1}$ is the $k$-fold divisor
function; i.e., the number of ways $n$ can be expressed as the product of $k$
natural numbers;

  * •

$\varphi(n):=n\prod_{j=1}^{r}(1-1/p_{j})$ is the Euler totient function;

  * •

$\mu(n)$ is the Möbius function, defined as $\mu(n)=(-1)^{r}$ if $n$ is
squarefree and $\mu(n)=0$ otherwise.

We write $f\ll g$, or $f=O(g)$, to indicate that in a range of $z$ (clear from
context), there exists a positive constant $C$ that depends only on $k$,
$\theta$, $\delta$, ${\rm diam}(\mathcal{H})$, $f_{1}$, $f_{2}$, $I_{1}$,
$I_{2}$, $A$, and $B$ such that $|f(z)|\leq C|g(z)|$. Also, the use of the
term “sufficiently large” may depend on $k$, $\theta$, $\delta$, ${\rm
diam}(\mathcal{H})$, $f_{1}$, $f_{2}$, $I_{1}$, $I_{2}$, $A$, and $B$. The
dependence of the implied constant on any additional parameters
$u_{1},\ldots,u_{n}$ is denoted by the symbol $\ll_{u_{1},\ldots,u_{n}}$. We
write $f\asymp g$ to denote that $f\ll g$ and $f\gg g$.

For $z\in\mathbb{C}$ let $e(z):=e^{2\pi iz}$. Given a locally integrable
function $f:\mathbb{C}\to\mathbb{C}$, for $\sigma\in\mathbb{R}$ we define

$\int_{(\sigma)}f(s)\,ds:=\lim_{T\to\infty}\int_{\sigma-
iT}^{\sigma+iT}f(s)\,ds$

provided that the limit exists.

For every $m\in\mathbb{N}\cup\\{0\\}$, we define the $m$-th Chebyshev
polynomial of the second kind $U_{m}(t)\in\mathbb{Z}[t]$ by the recurrence
relation

$U_{0}(t):=1,\qquad U_{1}(t):=2t,\qquad U_{m+2}(t):=2tU_{m+1}(t)-U_{m}(t).$

In particular, we have that the Chebyshev polynomials satisfy

$U_{m}(\cos\theta)=\sum_{j=0}^{m}e^{i(m-2j)\theta},\qquad\theta\in\mathbb{R}.$

##  3\. A Minorizing Fourier Expansion

In order to prove bounded gaps in the prime sets
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$, one must be able to detect when
$(\theta_{1}(p),\theta_{2}(p))\in I_{1}\times I_{2}$. We proceed as in Rouse
and Thorner [39, Lemma 3.1].

######  Lemma 3.1.

Let $I_{1},I_{2}\subseteq[0,\pi]$ be subintervals. Given
$M_{1},M_{2}\in\mathbb{N}$, there exists a real trigonometric polynomial of
the form

(3.1)
$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2})=\sum_{0\leq
m_{1}\leq M_{1}}\sum_{0\leq m_{2}\leq
M_{2}}\widehat{\mathcal{F}}_{I_{1},I_{2},M_{1},M_{2}}(m_{1},m_{2})U_{m_{1}}(\cos\theta_{1})U_{m_{2}}(\cos\theta_{2})$

with the following properties.

  1. (1)

If $\theta_{1},\theta_{2}\in[0,\pi]$, then
$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2})\leq\mathbbm{1}_{I_{1}\times
I_{2}}(\theta_{1},\theta_{2})$.

  2. (2)

The inequality

$|\widehat{\mathcal{F}}_{I_{1},I_{2},M_{1},M_{2}}(0,0)-\mu_{{\rm
ST}}(I_{1})\mu_{{\rm ST}}(I_{2})|\leq 8\Big{(}\frac{\mu_{\rm
ST}(I_{2})^{1/3}}{M_{1}+1}+\frac{\mu_{\rm
ST}(I_{1})^{1/3}}{M_{2}+1}+\frac{6}{(M_{1}+1)(M_{2}+1)}\Big{)}$

holds.

###### Proof.

We begin with some notation. Let $I_{1}=[a_{1},b_{1}]$, $I_{2}=[a_{2},b_{2}]$,
and $J_{1}:=[a_{1}/(2\pi),b_{1}/(2\pi)]$,
$J_{2}:=[a_{2}/(2\pi),b_{2}/(2\pi)]$. Given an interval $J\subseteq[0,1]$, the
Fourier expansion of the indicator function $\mathbbm{1}_{J}$ is given by

$\mathbbm{1}_{J}(\theta)=\sum_{n\in\mathbb{Z}}\widehat{\mathbbm{1}}_{J}(n)e(n\theta),\qquad\widehat{\mathbbm{1}}_{J}(n)=\int_{J}e(-nt)\,dt.$

Cochrane [8] proved that for $M_{1},M_{2}\in\mathbb{N}$, there exists a
trigonometric polynomial

$S(x_{1},x_{2})=\sum_{\begin{subarray}{c}0\leq|m_{1}|\leq M_{1}\\\
0\leq|m_{2}|\leq M_{2}\end{subarray}}\widehat{S}(m_{1},m_{2})\cdot
e(m_{1}x_{1}+m_{2}x_{2})$

with the following properties:

  1. (1)

for all $(x_{1},x_{2})\in[0,1]^{2}$,
$S(x_{1},x_{2})\leq\mathbbm{1}_{J_{1}\times J_{2}}(x_{1},x_{2})$;

  2. (2)

for all $(m_{1},m_{2})$, we have that
$\widehat{S}(m_{1},m_{2})+\widehat{S}(-m_{1},-m_{2})=2\operatorname{Re}\widehat{S}(m_{1},m_{2})$;

  3. (3)

for all $(m_{1},m_{2})$, we have that

(3.2)
$\displaystyle\Big{|}\widehat{\mathbbm{1}}_{J_{1}}(m_{1})\widehat{\mathbbm{1}}_{J_{2}}(m_{2})-\widehat{S}(m_{1},m_{2})\Big{|}\leq\frac{|J_{2}|}{M_{1}+1}+\frac{|J_{1}|}{M_{2}+1}+\frac{3}{(M_{1}+1)(M_{2}+1)}.$

If we define $U_{-1}(\cos\theta)=0$ and $U_{-2}(\cos\theta)=-1$, then

(3.3)
$\displaystyle\cos(n\theta)=\frac{1}{2}(U_{n}(\cos\theta)-U_{n-2}(\cos\theta)),\hskip
28.45274ptn\geq 0.$

We use these identities to obtain the Chebyshev polynomial approximation

$\displaystyle\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2})=S\Big{(}\frac{\theta_{1}}{2\pi},\frac{\theta_{2}}{2\pi}\Big{)}+S\Big{(}-\frac{\theta_{1}}{2\pi},\frac{\theta_{2}}{2\pi}\Big{)}+S\Big{(}\frac{\theta_{1}}{2\pi},-\frac{\theta_{2}}{2\pi}\Big{)}+S\Big{(}-\frac{\theta_{1}}{2\pi},-\frac{\theta_{2}}{2\pi}\Big{)}$

for $\mathbbm{1}_{I_{1}\times I_{2}}(\theta_{1},\theta_{2})$. Notice that
$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}$ is real-valued. We see for
$(\theta_{1},\theta_{2})\in(0,\pi)^{2}$ that

$\Big{(}-\frac{\theta_{1}}{2\pi},\frac{\theta_{2}}{2\pi}\Big{)},\Big{(}\frac{\theta_{1}}{2\pi},-\frac{\theta_{2}}{2\pi}\Big{)},\Big{(}-\frac{\theta_{1}}{2\pi},-\frac{\theta_{2}}{2\pi}\Big{)}\notin(J_{1}\times
J_{2})\ (\mathrm{mod}\ 1).$

Therefore,

(3.4)
$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2})\leq\mathbbm{1}_{J_{1}\times
J_{2}}\Big{(}\frac{\theta_{1}}{2\pi},\frac{\theta_{2}}{2\pi}\Big{)}=\mathbbm{1}_{I_{1}\times
I_{2}}(\theta_{1},\theta_{2}).$

By continuity of $\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}$, the inequality 3.4
holds for all $(\theta_{1},\theta_{2})\in[0,\pi]^{2}$.

We now write $\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}$ in terms of the Chebyshev
polynomials. First, we observe that

$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2})=4\widehat{S}(0,0)+4\sum_{\begin{subarray}{c}|m_{1}|\leq
M_{1}\\\ |m_{2}|\leq M_{2}\\\
\mathbf{m}\neq\mathbf{0}\end{subarray}}\widehat{S}(m_{1},m_{2})\cos(|m_{1}|\theta_{1})\cos(|m_{2}|\theta_{2}).$

By 3.3, this equals

$\displaystyle 4$
$\displaystyle\widehat{S}(0,0)+\sum_{\begin{subarray}{c}|m_{1}|\leq M_{1}\\\
|m_{2}|\leq M_{2}\\\
\mathbf{m}\neq\mathbf{0}\end{subarray}}\widehat{S}(m_{1},m_{2})\big{(}U_{|m_{1}|}(\cos\theta_{1})-U_{|m_{1}|-2}(\cos\theta_{1})\big{)}\big{(}U_{|m_{2}|}(\cos\theta_{2})-U_{|m_{2}|-2}(\cos\theta_{2})\big{)}.$

We compute

$\widehat{\mathcal{F}}_{I_{1},I_{2},M_{1},M_{2}}(0,0)=4\operatorname{Re}(\widehat{S}(0,0)-\widehat{S}(2,0)-\widehat{S}(0,2)+\widehat{S}(2,2))$

and

$\mu_{\rm
ST}([a,b])=2\operatorname{Re}\Big{(}\widehat{\mathbbm{1}}_{[\frac{a}{2\pi},\frac{b}{2\pi}]}(0)-\widehat{\mathbbm{1}}_{[\frac{a}{2\pi},\frac{b}{2\pi}]}(2)\Big{)}.$

Therefore, it follows from Item 3 for $S$ that

$\displaystyle|\widehat{\mathcal{F}}_{I_{1},I_{2},M_{1},M_{2}}(0,0)-\mu_{\rm
ST}(I_{1})\mu_{\rm ST}(I_{2})|$
$\displaystyle=4|\operatorname{Re}(\widehat{S}(0,0)-\widehat{\mathbbm{1}}_{J_{1}}(0)\widehat{\mathbbm{1}}_{J_{2}}(0)-\widehat{S}(2,0)+\widehat{\mathbbm{1}}_{J_{1}}(2)\widehat{\mathbbm{1}}_{J_{2}}(0)$
$\displaystyle-\widehat{S}(0,2)+\widehat{\mathbbm{1}}_{J_{1}}(0)\widehat{\mathbbm{1}}_{J_{2}}(2)+\widehat{S}(2,2)-\widehat{\mathbbm{1}}_{J_{1}}(2)\widehat{\mathbbm{1}}(2))|$
$\displaystyle\leq
16\Big{(}\frac{|J_{2}|}{M_{1}+1}+\frac{|J_{1}|}{M_{2}+1}+\frac{3}{(M_{1}+1)(M_{2}+1)}\Big{)}.$

We claim that if $j\in\\{1,2\\}$, then $|I_{j}|\leq\pi\mu_{\rm
ST}(I_{j})^{1/3}$. Indeed, writing $I_{j}=[x,x+y]\subseteq[0,\pi]$, we compute

$\pi\mu_{\rm
ST}([x,x+y])=2\int_{x}^{x+y}\sin^{2}\theta\,d\theta=y-\sin(x+y)\cos(x+y)+\sin(x)\cos(x).$

Taking first and second partial derivatives with respect to $x$ for $0\leq
x\leq\pi-y$, one finds the function on the right is minimized (for fixed $y$)
when $x=0$ (or $x=\pi-y$). So without loss of generality, we may let $x=0$.
Now, it suffices to show

$V(y):=2\pi^{2}\int_{0}^{y}\sin^{2}(\theta)\,d\theta-y^{3}\geq 0,\qquad 0\leq
y\leq\pi.$

The critical points of $V(y)$ are $y=0,\pi,2.148\ldots$ and one computes
$V(0)=V(\pi)=0$ and $V(2.148\ldots)=15.80\ldots$, therefore $V(y)\geq 0$ in
the interval $[0,\pi]$. Since $|J_{i}|=|I_{i}|/(2\pi)$ and
$|I_{i}|\leq\pi\mu_{\rm ST}(I_{i})^{1/3}$, Item 2 follows for
$\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}$. ∎

Given subintervals $I_{1},I_{2}\subseteq[0,\pi]$, we choose

(3.5) $M_{1}=\big{\lceil}27.8\cdot\mu_{\rm ST}(I_{1})^{-1}\mu_{\rm
ST}(I_{2})^{-2/3}-1\big{\rceil},\qquad M_{2}=\big{\lceil}27.8\cdot\mu_{\rm
ST}(I_{1})^{-2/3}\mu_{\rm ST}(I_{2})^{-1}-1\big{\rceil}.$

and define

(3.6)
$F_{I_{1},I_{2}}(\theta_{1},\theta_{2})=\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}(\theta_{1},\theta_{2}).$

######  Corollary 3.2.

In the notation of 3.5 and 3.6, the following properties hold:

  1. (1)

if $(\theta_{1},\theta_{2})\in[0,\pi]^{2}$, then
$F_{I_{1},I_{2}}(\theta_{1},\theta_{2})\leq\mathbbm{1}_{I_{1}\times
I_{2}}(\theta_{1},\theta_{2})$;

  2. (2)

$\widehat{F}_{I_{1},I_{2}}(0,0)>0$; and

  3. (3)

$(2(M_{1}+1)(M_{2}+1)+1)/\widehat{F}_{I_{1},I_{2}}(0,0)\leq 4581(\mu_{\rm
ST}(I_{1})\mu_{\rm ST}(I_{2}))^{-8/3}$.

###### Proof.

This follows from Item 2 in Lemma 3.1. ∎

We now state our main technical result, a Bombieri–Vinogradov type theorem for
trigonometric approximations, Theorem 3.3, whose proof we postpone until
Section 9. We use Theorem 3.3 and Corollary 3.2 in Section 4 to prove Theorem
1.1.

######  Theorem 3.3.

Let $f_{1}\not\sim f_{2}$ be non-CM holomorphic newforms with trivial
nebentypus and levels $N_{1},N_{2}$. Let $I_{1},I_{2}\subseteq[0,\pi]$ be
subintervals of positive length. Let
$\mathcal{F}=\mathcal{F}_{I_{1},I_{2},M_{1},M_{2}}$ be a trigonometric
polynomial as in (3.1). Then, for all
$\theta\in\Big{(}0,\frac{2}{2(M_{1}+1)(M_{2}+1)+1}\Big{)}$ and $A>0$,

$\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
(q,N_{1}N_{2})=1\end{subarray}}\max_{(a,q)=1}\Big{|}\sum_{\begin{subarray}{c}x<p\leq
2x\\\ p\equiv a\ (\mathrm{mod}\
q)\end{subarray}}\mathcal{F}(\theta_{1}(p),\theta_{2}(p))-\widehat{\mathcal{F}}(0,0)\frac{\pi(2x)-\pi(x)}{\varphi(q)}\Big{|}\ll\frac{x}{(\log
x)^{A}},$

where the implied constant only depends on $f_{1},f_{2},M_{1},M_{2},\theta,A$.

##  4\. Overview of Proof

In this section, we give an overview of the proof of Theorem 1.1 as well as
its common generalization with the Green–Tao theorem.

###  4.1. Modified Maynard–Tao Sieve

We present the necessary sieve theory for our proofs, following Maynard [29],
and Vatwani and Wong [45]. Let $F_{I_{1},I_{2}}$ be the minorizing polynomial
given in Corollary 3.2 with degrees $M_{1},M_{2}$. We define several
parameters which may depend on a uniform choice of $x\geq 16$:

$\widetilde{\theta}:=\frac{1}{(M_{1}+1)(M_{2}+1)+2},\qquad
0<\theta<\widetilde{\theta},\qquad\delta\in(0,\theta/2),$
$R:=x^{\theta/2-\delta},\qquad D_{0}>0,\qquad W:=\prod_{p\leq D_{0}}p,\qquad
U:=\prod_{\begin{subarray}{c}p\leq D_{0}\\\ p\nmid
N_{1}N_{2}\end{subarray}}p.$

Let $\mathcal{H}=\\{h_{1},\ldots,h_{k}\\}$ be an admissible set. For all
primes $p$ there exists an integer $a_{p}$ such that $a_{p}\not\equiv h_{j}\
(\mathrm{mod}\ p)$ for all $1\leq j\leq k$. By the Chinese Remainder theorem,
we can choose $u_{0}\ (\mathrm{mod}\ U)$ such that $u_{0}\equiv-a_{p}\mod p$
for all $p\mid U$. This way, we have that $u_{0}+h_{j}\not\equiv 0\
(\mathrm{mod}\ p)$ for all $p\mid U$ and $1\leq j\leq k$. Therefore, we deduce
that $(u_{0}+h_{j},U)=1$ for all $1\leq j\leq k$. For every $k$-tuple of
natural numbers $d_{1},\ldots,d_{k}$, we introduce a parameter

$\lambda_{d_{1},\ldots,d_{k}}\in\mathbb{R}$

that satisfies the following support condition. The parameter
$\lambda_{d_{1},\ldots,d_{k}}$ is nonzero only if $d:=\prod_{j=1}^{k}d_{j}$
satisfies $d<R$, $\mu^{2}(d)=1$, and we also have that $(d_{j},W)=1$ for all
$1\leq i\leq k$. Notice that the condition $\mu^{2}(d)=1$ implies that the
$d_{j}$ are squarefree and pairwise coprime. We define for each
$n\in\mathbb{N}$ a nonnegative weight

$w_{n}:=\begin{cases}\Big{(}\sum\limits_{d_{j}\mid n+h_{j}\,\forall
j}\lambda_{d_{1},\ldots,d_{k}}\Big{)}^{2}&\mbox{if $n\equiv u_{0}\
(\mathrm{mod}\ U)$,}\\\ 0&\mbox{otherwise.}\end{cases}$

We also define the sums

$\displaystyle S_{1}$ $\displaystyle:=\sum_{\begin{subarray}{c}x<n\leq 2x\\\
n\equiv u_{0}\ (\mathrm{mod}\ U)\end{subarray}}w_{n},$ $\displaystyle S_{2}$
$\displaystyle:=\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv u_{0}\
(\mathrm{mod}\
U)\end{subarray}}\Big{(}\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{j}),\cos\theta_{2}(n+h_{j}))\Big{)}w_{n},$
$\displaystyle S(x,\rho)$ $\displaystyle:=S_{2}-\rho S_{1}.$

Although $\theta_{j}(p)$ is defined only for $p$ prime, by abuse of notation
this is acceptable because the indicator function
$\mathbbm{1}_{\mathbb{P}}(n+h_{j})$ is non-vanishing if and only if $n+h_{j}$
is prime. Note that because $F_{I_{1},I_{2}}$ satisfies Item 1, we have that

$\displaystyle\sum_{x<n\leq
2x}\Big{(}\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})\mathbbm{1}_{I_{1}\times
I_{2}}(\theta_{1}(n+h_{j}),\theta_{2}(n+h_{j}))-\rho\Big{)}w_{n}\geq
S(x,\rho).$

Thus, it suffices to show that $S(x,\rho)>0$ for all $x$ sufficiently large in
order to deduce by the pigeonhole principle that there are infinitely many
integers $n\in\mathbb{N}$ such that the set $\\{n+h_{1},\ldots,n+h_{k}\\}$
contains at least $\lfloor\rho+1\rfloor$ primes which lie in
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$.

###  4.2. Bounded Gaps

In this section, we prove Theorem 1.1 using three propositions that are
analogues of [29, Propositions 4.1, 4.2, 4.3]. We postpone the proof of these
propositions in Section 5. While Theorem 1.1 follows from the second and third
propositions, we state the first proposition here as well for completeness.

The first proposition expresses the asymptotics for the sums $S_{1}$ and
$S_{2}$ in terms of the integral operators of a Riemann-integrable function
$G:[0,1]^{k}\to\mathbb{R}$, defined by

$\displaystyle I_{k}(G)$
$\displaystyle:=\int_{0}^{1}\cdots\int_{0}^{1}G(t_{1},\ldots,t_{k})^{2}\,dt_{1}\cdots
dt_{k},$ $\displaystyle J_{k}^{(m)}(G)$
$\displaystyle:=\int_{0}^{1}\cdots\int_{0}^{1}\Big{(}\int_{0}^{1}G(t_{1},\ldots,t_{k})\,dt_{m}\Big{)}^{2}dt_{1}\cdots
dt_{m-1}dt_{m+1}\cdots d_{k}.$

######  Proposition 4.1.

Let $\theta\in\Big{(}0,\frac{1}{(M_{1}+1)(M_{2}+1)+2}\Big{)}$. Let $D_{0}$ be
sufficiently large. Let $G:[0,1]^{k}\to\mathbb{R}$ be a smooth function
supported on the simplex
$\\{(t_{1},\ldots,t_{k})\in[0,1]^{k}:\sum_{j=1}^{k}t_{j}\leq 1\\}$, and define

$\lambda_{d_{1},\ldots,d_{k}}:=\Big{(}\prod_{j=1}^{k}\mu(d_{j})d_{j}\Big{)}\sum_{\begin{subarray}{c}r_{1},\ldots,r_{k}\\\
d_{j}\mid r_{j}\\\
(r_{j},W)=1\end{subarray}}\frac{\mu(\prod_{j=1}^{k}r_{j})^{2}}{\prod_{j=1}^{k}\varphi(r_{j})}G\Big{(}\frac{\log
r_{1}}{\log R},\ldots,\frac{\log r_{k}}{\log R}\Big{)}.$

In particular, $\lambda_{d_{1},\ldots,d_{k}}=0$ unless $\prod_{j=1}^{k}d_{j}$
is at most $R$, coprime to $W$, and squarefree. Then, we have that

(4.1) $\displaystyle S_{1}$
$\displaystyle=\operatorname{rad}(N_{1}N_{2})\cdot\frac{(1+O(1/D_{0}))\varphi(W)^{k}x(\log
R)^{k}}{W^{k+1}}I_{k}(G),$ (4.2) $\displaystyle S_{2}$
$\displaystyle=\varphi(\operatorname{rad}(N_{1}N_{2}))\cdot\widehat{F}_{I_{1},I_{2}}(0,0)\cdot\frac{(1+O(1/D_{0}))\varphi(W)^{k}x(\log
R)^{k+1}}{W^{k+1}\log x}\sum_{m=1}^{k}J_{k}^{(m)}(G).$

######  Remark 4.2.

In Maynard’s work [29], he defines $D_{0}$ as $\log\log\log x$. It was noticed
by Pintz [35] that Proposition 4.1 holds for any choice of $D_{0}$
sufficiently large. Indeed, Maynard’s estimates [29, Lemmata 6.2, 6.3] are
strong enough for this purpose.

The second proposition allows us to convert the ratio of integral operators
into a statement about the existence of primes in bounded intervals.

######  Proposition 4.3.

Let $\mathcal{H}=\\{h_{1},\ldots,h_{k}\\}$ be an admissible set and
$\mathcal{S}_{k}$ denote the set of all Riemann-integrable real functions
supported on $\\{(t_{1},\ldots,t_{k})\in[0,1]^{k}:\sum_{j=1}^{k}t_{j}\leq
1\\}$. Fix $\theta\in(0,\widetilde{\theta})$, and define

(4.3)
$\mathcal{M}_{k}:=\sup_{G\in\mathcal{S}_{k}}\frac{\sum_{m=1}^{k}J_{k}^{(m)}(G)}{I_{k}(G)},\hskip
28.45274ptr_{k}:=\Big{\lceil}\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)\mathcal{M}_{k}\frac{\theta}{2}\Big{\rceil}.$

Then, there are infinitely many integers $n$ such that at least $r_{k}$ of the
numbers $n+h_{j}$ lie in $\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$. In
particular, letting $P_{n}$ be the $n$-th prime in
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$, we have that

$\liminf\limits_{n\to\infty}(P_{n+r_{k}-1}-P_{n})\leq{\rm diam}(\mathcal{H}).$

The third proposition, which was proven in Polymath Research Project 8b [36,
Theorem 3.9], provides a lower bound on the value $\mathcal{M}_{k}$.

######  Proposition 4.4.

There exists an absolute and effectively computable constant $c_{1}>0$ such
that if $\mathcal{M}_{k}$ is the quantity defined in (4.3) and $k\geq c_{1}$,
Then

$\mathcal{M}_{k}\geq\log k-c_{1}.$

###### Proof.

See [36, Theorem 3.9]. ∎

We now use Propositions 4.3 and 4.4 to prove a more explicit version of
Theorem 1.1. The proof of Theorem 1.1 then follows.

######  Theorem 4.5.

Let $I_{1},I_{2}\subseteq[0,\pi]$ be proper subintervals with positive
Lebesgue measure. Letting $P_{n}$ be the $n$-th prime in
$\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$, we have that for $m\in\mathbb{N}$,

$\liminf_{n\to\infty}(P_{n+m}-P_{n})\ll\frac{N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}\exp\Big{(}\frac{2N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}\Big{)},$

where the implied constant is absolute and effective.

###### Proof.

We define

$k:=\max\Big{(}3,\Big{\lceil}\exp\Big{(}\frac{2N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}+c_{1}+1\Big{)}\Big{\rceil}\Big{)},\qquad\theta:=\Big{(}1-\frac{1}{\log
k}\Big{)}\widetilde{\theta}.$

Consider $r_{k}$ as in (4.3). Using Proposition 4.4, we have that

$r_{k}\geq\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)(\log
k-c_{1})\Big{(}1-\frac{1}{\log
k}\Big{)}\frac{\widetilde{\theta}}{2}>\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\frac{\widetilde{\theta}}{2}\widehat{F}_{I_{1},I_{2}}(0,0)(\log
k-c_{1}-1)\geq m.$

Thus, by setting $\mathcal{H}:=\\{p_{\pi(k)+1},\ldots,p_{\pi(k)+k}\\}$ we have
that, by the prime number theorem,

$\displaystyle{\rm diam}(\mathcal{H})\ll k\log
k\ll\frac{N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}\exp\Big{(}\frac{2N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}\Big{)},$

where the implied constants are absolute and effective. By Proposition 4.3,
the proof is complete. ∎

###### Proof of Theorem 1.1.

This follows from Theorem 4.5 using Corollary 3.2. ∎

###  4.3. Arithmetic Progressions

For every positive constant $\gamma>0$, we define the set of numbers

$\mathcal{R}(\mathcal{H},\gamma):=\bigg{\\{}n\in\mathbb{N}:P^{-}\Big{(}\prod_{j=1}^{k}(n+h_{j})\Big{)}\geq
n^{\gamma}\bigg{\\}}.$

We begin by stating a theorem of Pintz [34], which provides sufficient
criteria for the existence of arithmetic progressions of arbitrary length.

######  Theorem 4.6 ([34, Theorem 5]).

Let $\mathcal{H}=\\{h_{1},\ldots,h_{k}\\}$ be an admissible set. Given a set
$\mathcal{A}\subseteq\mathbb{Z}$, suppose there exist constants
$\gamma_{1},\gamma_{2}>0$ such that

(4.4)
$\mathcal{A}\subseteq\mathcal{R}(\mathcal{H},\gamma_{1}),\qquad\\#\\{n\leq
x:n\in\mathcal{A}\\}\geq\gamma_{2}\frac{x}{(\log x)^{k}}\,\emph{ for
all}x\emph{ sufficiently large}.$

Then, $\mathcal{A}$ contains infinitely many arithmetic progressions of
arbitrary length.

We have the following proposition whose proof we present in Section 6.

######  Proposition 4.7.

Consider the set

(4.5)
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1}):=\Big{\\{}n\in\mathbb{N}:\\#\Big{(}(n+\mathcal{H})\cap\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}\Big{)}\geq
m+1\Big{\\}}\cap\mathcal{R}(\mathcal{H},\gamma_{1}).$

There exists a constant $c_{2}>0$ such that if $m\in\mathbb{N}$ satisfies

(4.6)
$\exp\Big{(}\frac{2N_{1}N_{2}m}{\varphi(N_{1}N_{2})\widehat{F}_{I_{1},I_{2}}(0,0)\widetilde{\theta}}\Big{)}\leq
c_{2}k,$

then there exists $\gamma_{1}>0$ sufficiently small so that
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ contains
infinitely many arithmetic progressions of arbitrary length.

Using Theorems 4.6 and 4.7, we prove the following theorem.

######  Theorem 4.8.

Let $I_{1},I_{2}$ be subintervals of $[0,\pi]$ with positive Lebesgue measure.
Let $\mathcal{H}=\\{h_{1},\ldots,h_{k}\\}$ be an admissible set. If
$m\in\mathbb{N}$ satisfies (4.6), then there exists an $(m+1)$-element subset
$\mathcal{H}^{\prime}\subseteq\mathcal{H}$ such that

$\\{n\in\mathbb{N}:n+\mathcal{H}^{\prime}\subseteq\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}\\}$

contains infinitely many arithmetic progressions of arbitrary length.

###### Proof.

We follow the argument of [15]. By Theorems 4.6 and 4.7, the set
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ contains
infinitely many $t$-term arithmetic progressions for each $t\in\mathbb{N}$.

Let $\mathfrak{Col}$ be the set of all $(m+1)$-subsets of $\mathcal{H}$. Let
$M$ be a positive integer we choose later. Because
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ contains
infinitely many $M$-term arithmetic progressions, we let
$\\{n_{1}<\cdots<n_{M}\\}\subseteq\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$
be one such progression. Notice that for every $n_{j}$, there exists an
$(m+1)$-subset $\mathcal{C}_{j}\in\mathfrak{Col}$ such that
$n_{j}+\mathcal{C}_{j}\in\mathcal{P}_{f_{1},f_{2},I_{1},I_{2}}$.

We view $\mathfrak{Col}$ as a finite set of colors and color the index $j$
with $\mathcal{C}_{j}$ for each $j=1,\ldots,M$. By van der Waerden’s theorem,
for $M=M(t,m,k)$ sufficiently large, there exists a subset of indices
$L\subseteq\\{1,\ldots,M\\}$ which form a $t$-term monochromatic progression.
In particular, it follows that the set $\\{n_{j}:j\in
J\\}\subseteq\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ is
a $t$-term monochromatic progression (coloring $n_{j}$ with $\mathcal{C}_{j}$
for each $j=1,\ldots,M$). Because
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ has infinitely
many $M$-term progressions, we deduce that
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ has infinitely
many $t$-term monochromatic progressions. Therefore, by the pigeonhole
principle, there exists an $(m+1)$-subset
$\mathcal{H^{\prime}}\subseteq\mathcal{H}$ that occurs as a monochromatic
coloring for a $t$-term progression infinitely many times. Then,
$\mathcal{H^{\prime}}$ satisfies the theorem as desired. ∎

###  4.4. Structure of the Paper

The remainder of our paper is dedicated to two primary objectives. The first
is to prove Propositions 4.1, 4.3 and 4.7, which we do in Sections 5 and 6.
The second is to prove Theorem 3.3, which we do in Section 7 through Section
9. Sections 7 and 8 provide the necessary background on the theory of
$L$-functions. We prove Theorem 3.3 in Section 9.

##  5\. Proof of Maynard–Tao Propositions

In this section we prove Propositions 4.1 and 4.3. Throughout, we often write
$\operatorname{\mathbf{d}}$ and $\operatorname{\mathbf{e}}$ to abbreviate
$(d_{1},\ldots,d_{k})$ and $(e_{1},\ldots,e_{k})$ respectively.

###### Proof of Proposition 4.1.

To handle $S_{1}$, we expand the square of $w_{n}$ and swap the order of
summation to obtain

(5.1)
$S_{1}=\sum_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}\sum_{\begin{subarray}{c}x<n\leq
2x\\\ n\equiv u_{0}\ (\mathrm{mod}\ U)\\\ [d_{j},e_{j}]\mid n+h_{j}\forall
j\end{subarray}}1.$

We claim that the summands of $S_{1}$ are nonzero only if
$W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise coprime. This is because
the only case that $W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are not pairwise
coprime is when $(d_{j},e_{\ell})>1$ for some $j\neq\ell$. However,
$(d_{j},e_{\ell})$ divides $h_{j}-h_{\ell}$ but is also coprime to $W$, and so
taking $D_{0}$ sufficiently large so that $|h_{j}-h_{\ell}|\leq D_{0}$, we
conclude that $(d_{j},e_{\ell})=1$.

Because we may assume that $W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise
coprime and $U\mid W$, we may also assume that
$U,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise coprime. Our conditions
for the index $n\in\mathbb{N}$ above are that $n\in(x,2x]$, $n\equiv u_{0}\
(\mathrm{mod}\ U)$, and $n+h_{j}\equiv 0\ (\mathrm{mod}\ [d_{j},e_{j}])$ for
all $j$. By applying the Chinese remainder theorem to simplify the $k+1$
congruence conditions that we have, we may replace them by the condition
$n\equiv a\ (\mathrm{mod}\ q)$, where $q:=U\prod_{j=1}^{k}[d_{j},e_{j}]$ and
$a$ is some residue class modulo $q$. Therefore, the inner sum in (5.1)
simplifies as $x/q+O(1)$, and we obtain

$S_{1}=\frac{x}{U}\sideset{}{{}^{\prime}}{\sum}_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}\frac{\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}}{\prod_{j=1}^{k}[d_{j},e_{j}]}+O\Big{(}\sideset{}{{}^{\prime}}{\sum}_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}|\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}|\Big{)},$

where $\sideset{}{{}^{\prime}}{\sum}$ denotes the restriction that
$W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise coprime. This gives the
same sum as in [29] by pulling out a factor of $W/U$. Note that
$W/U=\operatorname{rad}(N_{1}N_{2})$ for $D_{0}$ sufficiently large.
Therefore, (4.1) follows by the identical argument presented in [29], using
Remark 4.2 to replace $o(1)$ with $O(1/D_{0})$.

We now handle the case of $S_{2}$. For each $m\in\\{1,\ldots,k\\}$ let

(5.2) $\displaystyle S_{2}^{(m)}:$
$\displaystyle=\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv u_{0}\
(\mathrm{mod}\
U)\end{subarray}}\mathbbm{1}_{\mathbb{P}}(n+h_{m})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{m}),\cos\theta_{2}(n+h_{m}))\Big{(}\sum_{d_{j}\mid
n+h_{j}\forall j}\lambda_{\operatorname{\mathbf{d}}}\Big{)}^{2}$ (5.3)
$\displaystyle=\sum_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}\sum_{\begin{subarray}{c}x<n\leq
2x\\\ n\equiv u_{0}\ (\mathrm{mod}\ U)\\\ [d_{j},e_{j}]\mid
n+h_{j}\end{subarray}}\mathbbm{1}_{\mathbb{P}}(n+h_{m})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{m}),\cos\theta_{2}(n+h_{m})).$

In particular, we have that $S_{2}=\sum_{m=1}^{k}S_{2}^{(m)}$. As we have
argued in the case of $S_{1}$, the summands of $S_{2}^{(m)}$ are nonzero only
if $W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise coprime, and the
argument presented for $S_{1}$ still holds. Now, let us notice that for our
inner sum in (5.3) we may assume

$n+h_{m}\text{ is prime},\qquad[d_{m},e_{m}]\mid
n+h_{m},\qquad(d_{m}e_{m},W)=1,\qquad\text{and}\quad d_{m}e_{m}\leq
R^{2}=x^{\theta/2-\delta}.$

If $[d_{m},e_{m}]\neq 1$, we have that $[d_{m},e_{m}]=n+h_{m}>x+h_{m}$. On the
other hand, $[d_{m},e_{m}]\leq x^{\theta/2-\delta}$ and $\theta/2-\delta<1$,
we arrive at a contradiction for all $x=x(\mathcal{H},\theta,\delta)$
sufficiently large. Therefore, we may assume that $[d_{m},e_{m}]=1$, in
particular $d_{m}=e_{m}=1$.

Similar to our argument for $S_{1}$, we may use the Chinese remainder theorem
applied to the $k+1$ congruence conditions modulo
$U,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ to write our inner sum in (5.3) as
taking place over $n\in(x,2x]$ and $n\equiv a\ (\mathrm{mod}\ q)$, where
$q:=U\prod_{j=1}^{k}[d_{j},e_{j}]$ and $a$ is some residue class modulo $q$.
We may assume that $n+h_{m}$ is a prime for some $n\in(x,2x]$, otherwise the
inner sum is zero. That is, we may assume there exists a prime $p=x+O({\rm
diam}(\mathcal{H}))$ that is congruent to $a+h_{m}\ (\mathrm{mod}\ q)$. Notice
that $(a+h_{m},q)$ divides $p$, and so either $(a+h_{m},q)=1$, or

$x+O({\rm diam}(\mathcal{H}))=p\leq q=O_{\delta}(x^{\theta/2}).$

By taking $x$ sufficiently large we obtain $p>q$, therefore $(a+h_{m},q)=1$.

We may write

(5.4)
$S_{2}^{(m)}=\sum_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}\sum_{\begin{subarray}{c}x<n\leq
2x\\\ n+h_{m}\equiv a\ (\mathrm{mod}\
q)\end{subarray}}\mathbbm{1}_{\mathbb{P}}(n+h_{m})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{m}),\cos\theta_{2}(n+h_{m})).$

Notice by shifting our index $n\in\mathbb{N}$ by $-h_{m}$ that the inner sum
is equal to

$\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv a\ (\mathrm{mod}\
q)\end{subarray}}\mathbbm{1}_{\mathbb{P}}(n)F_{I_{1},I_{2}}(\cos\theta_{1}(n),\cos\theta_{2}(n))+O({\rm
diam}(\mathcal{H})).$

Because the inner sum index is counting primes in residue classes, it is
natural that we want to consider the error term

$E(x,q):=1+\max_{(a,q)=1}\Big{|}\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv
a\ (\mathrm{mod}\
q)\end{subarray}}\mathbbm{1}_{\mathbb{P}}(n)F_{I_{1},I_{2}}(\cos\theta_{1}(n),\cos\theta_{2}(n))-\widehat{F}_{I_{1},I_{2}}(0,0)\frac{\pi(2x)-\pi(x)}{\varphi(q)}\Big{|}.$

By shifting indices and isolating a main term from the error term $E(x,q)$, we
are able to write the inner sum in (5.4) as

(5.5)
$\displaystyle\widehat{F}_{I_{1},I_{2}}(0,0)\frac{\pi(2x)-\pi(x)}{\varphi(q)}+O(E(x,q)).$

Substituting (5.5) for the inner sum of (5.4) and using the multiplicativity
of $\varphi$ to pull out a factor of $\varphi(U)$ from $\varphi(q)$, we obtain

$S_{2}^{(m)}=\widehat{F}_{I_{1},I_{2}}(0,0)\frac{\pi(2x)-\pi(x)}{\varphi(U)}\sideset{}{{}^{\prime}}{\sum}_{\begin{subarray}{c}\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}\\\
d_{m}=1\\\
e_{m}=1\end{subarray}}\frac{\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}}{\prod_{j=1}^{k}\varphi([d_{j},e_{j}])}+O\Big{(}\sideset{}{{}^{\prime}}{\sum}_{\operatorname{\mathbf{d}},\operatorname{\mathbf{e}}}|\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}|E(x,q)\Big{)},$

where $\small\sideset{}{{}^{\prime}}{\sum}$ denotes the restriction that
$W,[d_{1},e_{1}],\ldots,[d_{k},e_{k}]$ are pairwise coprime.

We now employ Theorem 3.3 to bound the error term. We have as in [29, (5.9)
and (6.3)] that $\lambda_{\max}\ll G_{\max}(\log R)^{k}$, where

$\lambda_{\max}:=\max_{d_{1},\ldots,d_{k}}|\lambda_{d_{1},\ldots,d_{k}}|$

and

$G_{\max}:=\sup_{(t_{1},\ldots,t_{k})\in[0,1]^{k}}|G(t_{1},\ldots,t_{k})|+\sum_{j=1}^{k}\Big{|}\frac{\partial
G}{\partial t_{j}}(t_{1},\ldots,t_{k})\Big{|}.$

This gives a bound for
$|\lambda_{\operatorname{\mathbf{d}}}\lambda_{\operatorname{\mathbf{e}}}|$. By
the support of $\lambda_{d_{1},\ldots,d_{k}}$, we can restrict the range of
possible $q$’s to squarefree $r<R^{2}W$ such that $(r,N_{1}N_{2})=1$. Now, for
any squarefree $r$, there are at most $\tau_{3k}(r)$ choices of
$d_{1},\ldots,d_{k},e_{1},\ldots,e_{k}$ such that
$r=U\prod_{j=1}^{k}[d_{j},e_{j}]$ where $\tau_{3k}(r)$ is the number of ways
to write $r$ as a product of $3k$ positive integers. Thus, we obtain an error
term of

$\displaystyle\ll y_{\max}^{2}(\log
R)^{2k}\sum_{\begin{subarray}{c}r<R^{2}W\\\
(r,N_{1}N_{2})=1\end{subarray}}\mu(r)^{2}\tau_{3k}(r)E(x,r).$

Applying Cauchy-Schwarz, we obtain an error that is

$\ll y_{\max}^{2}(\log
R)^{2k}\Big{(}\sum_{r<R^{2}W}\mu(r)^{2}\tau_{3k}^{2}(r)E(x,r)\Big{)}^{1/2}\Big{(}\sum_{\begin{subarray}{c}r<R^{2}W\\\
(r,N_{1}N_{2})=1\end{subarray}}\mu(r)^{2}E(x,r)\Big{)}^{1/2}.$

Notice that for $E(x,r)$ we have that

$E(x,r)\ll_{F_{I_{1},I_{2}}}\max_{(a,q)=1}\frac{\pi(2x;r,a)-\pi(x;r,a)}{\varphi(r)}+\frac{\pi(2x)-\pi(x)}{\varphi(r)}\ll\frac{x}{\varphi(r)}.$

Therefore, the middle factor is $O(x^{1/2}(\log x)^{3k/2})$. On the other
hand, note that $R^{2}W\leq x^{\theta}$ for $x=x(\delta)$ sufficiently large,
so we can apply Theorem 3.3 to bound the second factor by
$O_{A}\Big{(}x^{1/2}(\log x)^{-A}\Big{)}$ for any $A>0$. So the total error is
$\ll_{A}y_{\max}^{2}x(\log x)^{-A}$. This gives the same shape of error as in
[29, (5.20)], so (4.2) follows by the same arguments as in [29], using Remark
4.2 to replace $o(1)$ with $O(1/D_{0})$. Also, the main term is adjusted by a
factor of $\varphi(W)/\varphi(U)$. This factor equals
$\varphi(\operatorname{rad}(N_{1}N_{2}))$ for $D_{0}$ sufficiently large. ∎

###### Proof of Proposition 4.3.

Recall that if we can show $S(x,\rho)>0$ for all large $x$, then there exist
infinitely many integers $n$ such that at least $\lfloor\rho+1\rfloor$ of the
$n+h_{j}$ are prime.

By definition of $\mathcal{M}_{k}$ and the density of smooth functions in the
space of Riemann-integrable functions, there exists a smooth function $G_{1}$
such that
$\sum_{j=1}^{k}J_{k}^{(m)}(G_{1})>(\mathcal{M}_{k}-2\delta)I_{k}(G_{1})>0$.
Applying Proposition 4.1 with $G=G_{1}$, we have that since $\log R/\log
x=\theta/2-\delta$,

$\displaystyle S(x,\rho)$ $\displaystyle=\frac{\varphi(W)^{k}x(\log
R)^{k}}{W^{k+1}}\Big{(}\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)\frac{\log
R}{\log x}\sum_{m=1}^{k}J_{k}^{(m)}(F)-\rho I_{k}(F)+o(1)\Big{)}$
$\displaystyle\geq\frac{\varphi(W)^{k}x(\log
R)^{k}I_{k}(F)}{W^{k+1}}\Big{(}\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)\Big{(}\frac{\theta}{2}-\delta\Big{)}(\mathcal{M}_{k}-\delta)-\rho+o(1)\Big{)}.$

If we let

$\rho=\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)\frac{\theta\mathcal{M}_{k}}{2}-\varepsilon,$

then by choosing $\delta=\delta(\varepsilon)$ small enough we see that
$S(x,\rho)>0$ for all large $x$. Since
$\lfloor\rho+1\rfloor=\big{\lceil}\frac{\varphi(N_{1}N_{2})}{N_{1}N_{2}}\widehat{F}_{I_{1},I_{2}}(0,0)\theta
M_{k}/2\big{\rceil}$ if $\varepsilon$ is small enough, we are done since at
least $\lfloor\rho+1\rfloor$ of the $n+h_{j}$ are prime for infinitely many
$n$. ∎

##  6\. Proof of Proposition 4.7

Recall that we are interested in the growth of the set
$\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$ defined in
(4.5). We first restrict the sums $S_{1}$ and $S_{2}$ to the indices
$n\in\mathbb{N}$ that lie in $\mathcal{R}(\mathcal{H},\gamma_{1})$. That is,
we define the sums

$\displaystyle S_{1}^{+}(\gamma_{1})$
$\displaystyle:=\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv u_{0}\
(\mathrm{mod}\ U)\\\
n\in\mathcal{R}(\mathcal{H},\gamma_{1})\end{subarray}}w_{n},$ $\displaystyle
S_{2}^{+}(\gamma_{1})$ $\displaystyle:=\sum_{\begin{subarray}{c}x<n\leq 2x\\\
n\equiv u_{0}\ (\mathrm{mod}\ U)\\\
n\in\mathcal{R}(\mathcal{H},\gamma_{1})\end{subarray}}\Big{(}\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})F_{I_{1},I_{2}}(\theta_{1}(n+h_{j}),\theta_{2}(n+h_{j}))\Big{)}w_{n}.$

We want to show for $\gamma_{1}>0$ sufficiently small that the sums
$S_{j}^{+}(\gamma_{1})$ differ negligibly from $S_{j}$. That is, we want to
study the differences

$S_{1}^{-}(\gamma_{1}):=S_{1}-S_{1}^{+}(\gamma_{1}),\qquad
S_{2}^{-}(\gamma_{1}):=S_{2}-S_{2}^{+}(\gamma_{1}).$

In fact, we will further decompose $S_{1}^{-}(\gamma_{1})$ into the sums

$S_{1,p}^{(j)}:=\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\equiv u_{0}\
(\mathrm{mod}\ U)\\\ p\mid n+h_{j}\end{subarray}}w_{n},\qquad 1\leq j\leq
k,\qquad p\text{ prime}.$

We have the following two lemmata due to Vatwani and Wong [45].

######  Lemma 6.1.

For any $j\in\\{1,\ldots,k\\}$ and any prime $p>D_{0}$ with $\frac{\log
p}{\log R}<\varepsilon$, one has for $\varepsilon>0$ sufficiently small that

$S_{1,p}^{(j)}\ll\frac{(\log p)^{2}}{p(\log R)^{2}}\frac{x(\log R)^{k}}{U}.$

###### Proof.

A proof is given by Vatwani and Wong [45, Lemma 5.1] when $U$ is equal to

$\prod_{\begin{subarray}{c}p\leq D_{0}\\\ p\,\nmid\,d_{K}\end{subarray}}p$

where $d_{K}$ is the discriminant of a number field $K$. By replacing $d_{K}$
with $N_{1}N_{2}$, we are done. ∎

######  Lemma 6.2.

For $\varepsilon>0$ sufficiently small, there exists
$\gamma_{1}=\gamma_{1}(\varepsilon)>0$ such that for $x=x(\varepsilon)$
sufficiently large,

$S_{1}^{-}(\gamma_{1})\leq\varepsilon\frac{x(\log
R)^{k}}{U}\qquad\text{and}\qquad
S_{2}^{-}(\gamma_{1})\leq\varepsilon\frac{x(\log R)^{k}}{U}.$

###### Proof.

The following is based on [45, Lemma 5.2]. Since $S_{1}^{-}(\gamma_{1})$
includes only values of $n$ for which $n+h_{j}$ has small prime factors for
each $i$, we have that

$S_{1}^{-}(\gamma_{1})\leq\sum_{j=1}^{k}\sum_{p\leq(2x)^{\gamma_{1}}}S_{1,p}^{(j)}.$

Then, we have that, for our indices $p$ (and $x\geq 2$), $\frac{\log p}{\log
R}\leq\gamma_{1}\frac{4}{\theta-2\delta}$. Therefore, taking
$\gamma_{1}\leq\frac{\theta-2\delta}{8}\varepsilon$ and applying Lemma 6.1,
there exists a constant $c_{3}>0$ such that

$S_{1}^{-}(\gamma_{1})\leq c_{3}\frac{x(\log
R)^{k}}{U}\sum_{p\leq(2x)^{\gamma_{1}}}\frac{(\log p)^{2}}{p(\log R)^{2}}.$

By the prime number theorem, one has that for $x=x(\gamma_{1})$ sufficiently
large,

$\sum_{p\leq(2x)^{\gamma_{1}}}\frac{(\log p)^{2}}{p}\ll\gamma_{1}^{2}(\log
x)^{2}.$

Therefore, noting that $\frac{\log x}{\log R}\ll 1$, there exists a constant
$c_{4}>0$ such that

$S_{1}^{-}(\gamma_{1})\leq c_{4}\gamma_{1}^{2}\frac{x(\log R)^{k}}{U}.$

We now choose $\gamma_{1}=\gamma_{1}(\varepsilon)$ sufficiently small so that
$c_{4}\gamma_{1}^{2}\leq\varepsilon$ and
$\gamma_{1}\leq\frac{\theta-2\delta}{8}\varepsilon$. The result follows. ∎

We are now ready to prove Proposition 4.7.

###### Proof of Proposition 4.7.

To prove an estimate for the growth of the set
$\mathcal{B}:=\mathcal{A}_{f_{1},f_{2},I_{1},I_{2}}(\mathcal{H},\gamma_{1})$,
we begin performing manipulations on the dyadic sum

$\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\in\mathcal{B}\end{subarray}}1$

that are analogous to [45, Proof of Theorem 5.5]. We first notice that for
$n\in\mathcal{B}$ and $\rho$ as in the proof of Proposition 4.3,

$0<\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{j}),\cos\theta_{2}(n+h_{j}))-\rho\leq
k-m\leq k.$

In particular, there exists a constant $c_{5}>0$ such that

$\sum_{\begin{subarray}{c}x<n\leq 2x\\\ n\in\mathcal{B}\end{subarray}}1\geq
c_{5}\sum_{\begin{subarray}{c}x<n\leq 2x\\\
n\in\mathcal{B}\end{subarray}}\Big{(}\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{j}),\cos\theta_{2}(n+h_{j}))-\rho\Big{)}.$

Notice that for $n\in\mathcal{R}(\mathcal{H},\gamma_{1})$,

(6.1) $\sum_{d_{j}\mid n+h_{j}\,\forall j}1\ll_{\gamma_{1}}1.$

Also, from [29, (5.9) and (6.3)], we have that

(6.2) $\lambda_{\max}\ll y_{\max}(\log R)^{k}\ll(\log R)^{k}.$

It follows from 6.1 and 6.2 that there exists a constant
$c_{6}=c_{6}(\gamma_{1})>0$ such that for all $k$-tuples $d_{1},\ldots,d_{k}$,

$c_{5}\geq\frac{c_{6}}{(\log R)^{2k}}\Big{(}\sum_{d_{j}\mid n+h_{j}\,\forall
j}\lambda_{d_{1},\ldots,d_{k}}\Big{)}^{2}=\frac{c_{6}}{(\log R)^{2k}}w_{n}.$

Therefore, we have that

$\displaystyle\sum_{\begin{subarray}{c}x<n\leq 2x\\\
n\in\mathcal{B}\end{subarray}}1$ $\displaystyle\geq\frac{c_{6}}{(\log
R)^{2k}}\sum_{\begin{subarray}{c}x<n\leq 2x\\\
n\in\mathcal{B}\end{subarray}}\Big{(}\sum_{j=1}^{k}\mathbbm{1}_{\mathbb{P}}(n+h_{j})F_{I_{1},I_{2}}(\cos\theta_{1}(n+h_{j}),\cos\theta_{2}(n+h_{j}))-\rho\Big{)}w_{n}$
$\displaystyle=\frac{c_{6}}{(\log R)^{2k}}\Big{(}S_{2}^{+}(\gamma_{1})-\rho
S_{1}^{+}(\gamma_{1})\Big{)}.$

By Proposition 4.1, if $D_{0}$ is sufficiently large, then

$S_{2}-\rho
S_{1}=x\Big{(}1+O\Big{(}\frac{1}{D_{0}}\Big{)}\Big{)}\frac{\varphi(W)^{k}(\log
R)^{k}}{W^{k+1}}\Big{(}\sum_{m=1}^{k}J_{k}^{(m)}(G)-\rho\frac{\log R}{\log
x}I_{k}(G)\Big{)}.$

Choose $m$ so that (4.6) holds. By following Propositions 4.3 and 4.5 with
$D_{0}$ sufficiently large but fixed with respect to $x$, there exists a
constant $c_{7}>0$ such that

$S_{2}-\rho S_{1}\geq 2c_{7}x(\log x)^{k}.$

Applying Lemma 6.2 with $\varepsilon$ sufficiently small so that
$S_{2}^{-}(\gamma_{1})+\rho S_{1}^{-}(\gamma_{1})\leq c_{7}x(\log x)^{k}$, it
follows that

$S_{2}^{+}(\gamma_{1})-\rho S_{1}^{+}(\gamma_{1})\geq c_{7}x(\log x)^{k}.$

Taking $\gamma_{2}=c_{6}c_{7}(\frac{\log x}{\log R})^{2k}$ and applying
Theorem 4.6 to the set $\mathcal{B}$ with constants $\gamma_{1}$ and
$\gamma_{2}$, this completes the proof. ∎

##  7\. Automorphic $L$-functions

In this section, we recall some of the theory of automorphic representations
and their associated $L$-functions.

###  7.1. Automorphic representations

Let $\mathbb{A}_{\mathbb{Q}}$ be the adele ring over $\mathbb{Q}$. For every
natural number $m\in\mathbb{N}$, let $\mathfrak{F}_{m}$ be the set of cuspidal
automorphic representations of
$\operatorname{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with trivial central
character. Let $\pi\in\mathfrak{F}_{m}$. There exist smooth admissible
representations $\pi_{\infty}$ of $\operatorname{GL}_{m}(\mathbb{R})$ and
$\pi_{p}$ of $\operatorname{GL}_{m}(\mathbb{Q}_{p})$ for every prime $p$ such
that $\pi$ equals the restricted tensor product

$\pi=\pi_{\infty}\otimes\Big{(}\sideset{}{{}^{\prime}}{\bigotimes}_{p}\pi_{p}\Big{)}.$

Moreover, $\pi_{p}$ is ramified for at most finitely many $p$. For every prime
$p$, there exist $m$ complex numbers $(\alpha_{j,\pi}(p))_{j=1}^{m}$ that form
the local $L$-function

$L(s,\pi_{p}):=\prod_{j=1}^{m}(1-\alpha_{j,\pi}(p)p^{-s})^{-1}.$

In particular, the $L$-function associated to $\pi$ has an absolutely
convergent Euler product

$L(s,\pi):=\prod_{p}L(s,\pi_{p})=\prod_{p}\prod_{j=1}^{m}(1-\alpha_{j,\pi}(p)p^{-s})^{-1}=\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s},\qquad\operatorname{Re}(s)>1.$

There exist $m$ complex Langlands parameters $(\kappa_{j,\pi})_{j=1}^{m}$ that
form the local $L$-function

$L(s,\pi_{\infty}):=\pi^{-ms/2}\prod_{j=1}^{m}\Gamma\Big{(}\frac{s+\kappa_{j,\pi}}{2}\Big{)}.$

There also exists a conductor $q_{\pi}\in\mathbb{N}$, and
$r_{\pi}\in\\{0,1\\}$ which is $1$ if and only if $\pi$ is the trivial
representation, such that the completed $L$-function

$\Lambda(s,\pi):=(s(1-s))^{r_{\pi}}q_{\pi}^{s/2}L(s,\pi_{\infty})L(s,\pi)$

is entire of order $1$. If $\widetilde{\pi}\in\mathfrak{F}_{m}$ is the dual of
$\pi$, then

$\\{\alpha_{j,\widetilde{\pi}}(p)\\}=\\{\overline{\alpha_{j,\pi}(p)}\\},\qquad\\{\kappa_{j,\widetilde{\pi}}\\}=\\{\overline{\kappa_{j,\pi}}\\},\qquad
q_{\widetilde{\pi}}=q_{\pi},\qquad r_{\widetilde{\pi}}=r_{\pi}.$

There exists $W(\pi)\in\mathbb{C}$ of modulus $1$ such that the functional
equation

$\Lambda(s,\pi)=W(\pi)\Lambda(1-s,\widetilde{\pi})$

holds for all $s\in\mathbb{C}$. The nontrivial zeros of $L(s,\pi)$ are the
zeros of $\Lambda(s,\pi)$, and they lie in the critical strip
$0\leq\operatorname{Re}(s)\leq 1$. Finally, the analytic conductor is defined
as

$C(s,\pi):=q_{\pi}\prod_{j=1}^{m}(|s+\kappa_{j,\pi}|+3),\qquad
C(\pi):=C(0,\pi).$

###  7.2. Rankin–Selberg $L$-functions

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
Let $r_{\pi\times\pi^{\prime}}=1$ if $\pi^{\prime}=\widetilde{\pi}$, and
otherwise $r_{\pi\times\pi^{\prime}}=0$. For every prime $p\nmid
q_{\pi}q_{\pi^{\prime}}$, we define the local $L$-function

$L(s,\pi_{p}\times\pi_{p}^{\prime}):=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}(1-\alpha_{j,\pi}(p)\alpha_{j^{\prime},\pi^{\prime}}(p)p^{-s})^{-1}.$

For each $p\mid q_{\pi}q_{\pi^{\prime}}$ and each pair of integers
$(j,j^{\prime})\in[1,m]\times[1,m^{\prime}]$, Jacquet, Piatetskii-Shapiro, and
Shalika [18] defined complex numbers
$\alpha_{j,j,^{\prime},\pi\times\pi^{\prime}}(p)$ and
$\kappa_{j,j^{\prime},\pi\times\pi^{\prime}}$ such that if

$L(s,\pi_{p}\times\pi_{p}^{\prime}):=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}(1-\alpha_{j,j^{\prime},\pi,\pi^{\prime}}(p)p^{-s})^{-1}$

and

$L(s,\pi_{\infty}\times\pi_{\infty}^{\prime}):=\pi^{-mm^{\prime}s/2}\prod_{j,j^{\prime}}\Gamma\Big{(}\frac{s+\kappa_{j,j^{\prime},\pi\times\pi^{\prime}}}{2}\Big{)},$

then the Rankin–Selberg $L$-function

$L(s,\pi\times\pi^{\prime}):=\prod_{p}L(s,\pi_{p}\times\pi_{p}^{\prime})=\sum_{n=1}^{\infty}\lambda_{\pi\times\pi^{\prime}}(n)n^{-s}$

converges absolutely for $\operatorname{Re}(s)>1$ and satisfies the following
theorem.

######  Theorem 7.1.

Let $(\pi,\pi^{\prime})\in\mathfrak{F}_{m}\times\mathfrak{F}_{m^{\prime}}$.
There exists a conductor $q_{\pi\times\pi^{\prime}}\in\mathbb{N}$ such that
the completed $L$-function

(7.1)
$\Lambda(s,\pi\times\pi^{\prime}):=(s(1-s))^{r_{\pi\times\pi^{\prime}}}q_{\pi\times\pi^{\prime}}^{s/2}L(s,\pi_{\infty}\times\pi_{\infty}^{\prime})L(s,\pi\times\pi^{\prime})$

is entire of order $1$. Moreover, there exists
$W(\pi\times\pi^{\prime})\in\mathbb{C}$ of modulus $1$ such that

$\Lambda(s,\pi\times\pi^{\prime})=W(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime}).$

###### Proof.

See [18]. ∎

######  Remark 7.2.

Moreover, [18] shows that $L(s,\pi\times\pi^{\prime})$ is holomorphic away
from a possible pole at $s=1$, which occurs if and only if
$\pi^{\prime}=\widetilde{\pi}$.

Let $\pi\in\mathfrak{F}_{m}$, $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$, and
$\chi$ be a primitive Dirichlet character modulo $q_{\chi}$. The nontrivial
zeros of $L(s,\pi\times(\pi^{\prime}\otimes\chi))$ are the zeros of
$\Lambda(s,\pi\times(\pi^{\prime}\otimes\chi))$, and they lie in the critical
strip $0\leq\operatorname{Re}(s)\leq 1$. For the conductor
$q_{\pi\times(\pi^{\prime}\otimes\chi)}$, Bushnell and Henniart [5] proved
that

(7.2) $q_{\pi\times(\pi^{\prime}\otimes\chi)}\mid
q_{\pi}^{m^{\prime}}q_{\pi^{\prime}}^{m}q_{\chi}^{mm^{\prime}}.$

Define

$C(s,\pi\times(\pi^{\prime}\otimes\chi)):=q_{\pi\times(\pi^{\prime}\otimes\chi)}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}(|s+\kappa_{j,j^{\prime},\pi\times\pi^{\prime}}|+3).\quad
C(\pi\times(\pi^{\prime}\otimes\chi)):=C(0,\pi\times(\pi^{\prime}\otimes\chi)),$

Brumley [23, Appendix] proved that

(7.3) $C(\pi\times(\pi^{\prime}\otimes\chi))\ll
C(\pi)^{m^{\prime}}C(\pi^{\prime})^{m}q_{\chi}^{mm^{\prime}},$

where the implied constant only depends on $m,m^{\prime}$. One can also check
that

(7.4) $C(it,\pi\times(\pi^{\prime}\otimes\chi))\leq
C(\pi\times(\pi^{\prime}\otimes\chi))(|t|+3)^{mm^{\prime}}.$

If $\chi$ is a primitive Dirichlet character, we denote the following
Dirichlet coefficients

$\displaystyle
L(s,\pi\times(\pi^{\prime}\otimes\chi))=\sum_{n=1}^{\infty}\lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)n^{-s},\qquad\frac{1}{L(s,\pi\times(\pi^{\prime}\otimes\chi))}=\sum_{n=1}^{\infty}\mu_{\pi\times(\pi^{\prime}\otimes\chi)}(n)n^{-s},$
$\displaystyle-\frac{L^{\prime}}{L}(s,\pi\times(\pi^{\prime}\otimes\chi))=\sum_{n=1}^{\infty}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)n^{-s}.$

Note, if $\chi$ is the trivial character then
$L(s,\pi\times(\pi^{\prime}\otimes\chi))=L(s,\pi\times\pi^{\prime})$ and if
$\pi^{\prime}$ is the trivial representation then
$L(s,\pi\times(\pi^{\prime}\otimes\chi))=L(s,\pi)$, so we are setting notation
for these $L$-functions as well. Using computations analogous to those from
Luo, Rudnick, and Sarnak [46, Lemma 2.1], we find for
$(q_{\pi}q_{\pi^{\prime}},q_{\chi})=1$ that
$\alpha_{j,\pi\times(\pi^{\prime}\otimes\chi)}(p)=\alpha_{j,\pi\times\pi^{\prime}}(p)\chi(p)$.
In particular, if $n\in\mathbb{N}$, then

$\displaystyle\lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)$
$\displaystyle=\lambda_{\pi\times\pi^{\prime}}(n)\chi(n),$ (7.5)
$\displaystyle\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)$
$\displaystyle=\Lambda_{\pi\times\pi^{\prime}}(n)\chi(n),$
$\displaystyle\mu_{\pi\times(\pi^{\prime}\otimes\chi)}(n)$
$\displaystyle=\mu_{\pi\times\pi^{\prime}}(n)\chi(n).$

We note that finding an upper bound on the absolute value of one of these
numbers is independent of $\chi$. Furthermore, there exist theorems which
allow decoupling of the dependencies on $\pi$ and $\pi^{\prime}$. For example,
Brumley [41, Appendix] proved that for all $n\in\mathbb{N}$,

(7.6)
$|\Lambda_{\pi\times\pi^{\prime}}(n)|\leq\sqrt{\Lambda_{\pi\times\widetilde{\pi}}(n)\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}.$

Also, Jiang, Lü, and Wang [25, Lemma 3.1] later proved that for all
$n\in\mathbb{N}$,

(7.7)
$|\lambda_{\pi\times\pi^{\prime}}(n)|\leq\sqrt{\lambda_{\pi\times\widetilde{\pi}}(n)\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(n)}.$

###  7.3. Isobaric sums

For $t\in\mathbb{R}$, consider the character $|\cdot|^{it}$, where $|\cdot|$
is the adelic norm. The $L$-function of $\pi\otimes|\cdot|^{it}$ is
$L(s+it,\pi)$. If $\ell\in\mathbb{N}$ and
$(\pi_{j},t_{j})\in\mathfrak{F}_{m_{j}}\times\mathbb{R}$ for all $1\leq
j\leq\ell$, then the isobaric sum
$\Pi=\pi_{1}\otimes|\cdot|^{it_{1}}\boxplus\cdots\boxplus\pi_{\ell}\otimes|\cdot|^{it_{\ell}}$
is an automorphic representation of
$\operatorname{GL}_{m_{1}+\cdots+m_{\ell}}(\mathbb{A}_{\mathbb{Q}})$ whose
$L$-function is

$L(s,\Pi)=\prod_{j=1}^{\ell}L(s+it_{j},\pi_{j}).$

We let $\mathfrak{A}_{m}$ denote the set of isobaric automorphic
representations of $\operatorname{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$.

We extend the Rankin–Selberg convolution to isobaric sums of cuspidal
automorphic representations. If
$\Pi=\pi_{1}\otimes|\cdot|^{it_{1}}\boxplus\cdots\boxplus\pi_{\ell}\otimes|\cdot|^{it_{\ell}}\in\mathfrak{A}_{m}$
and
$\Pi^{\prime}=\pi^{\prime}_{1}\otimes|\cdot|^{it^{\prime}_{1}}\boxplus\cdots\boxplus\pi^{\prime}_{\ell^{\prime}}\otimes|\cdot|^{it^{\prime}_{\ell^{\prime}}}\in\mathfrak{A}_{m^{\prime}}$,
then their Rankin–Selberg convolution is

(7.8)
$L(s,\Pi\times\Pi^{\prime}):=\prod_{j=1}^{\ell}\prod_{j^{\prime}=1}^{\ell^{\prime}}L(s+it_{j}+it^{\prime}_{j^{\prime}},\pi_{j}\times\pi^{\prime}_{j^{\prime}}).$

######  Lemma 7.3.

Let $\Pi$ be an isobaric sum of cuspidal automorphic representations. Then the
Dirichlet series for $L(s,\Pi\times\widetilde{\Pi})$, $\log
L(s,\Pi\times\widetilde{\Pi})$, and
$-\frac{L^{\prime}}{L}(s,\Pi\times\widetilde{\Pi})$ with
$\operatorname{Re}(s)>1$ have nonnegative Dirichlet coefficients.

###### Proof.

See [22, Lemma a]. ∎

###  7.4. $L$-function Estimates in the Critical Strip

Given $\pi\in\mathfrak{F}_{m}$, $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$,
and $\chi$ a primitive Dirichlet character modulo $q_{\chi}$, we prove
estimates for the auxiliary function

$\mathcal{L}(s,\pi\times(\pi^{\prime}\otimes\chi)):=\Big{(}\frac{s-1}{s+1}\Big{)}^{r_{\pi\times(\pi^{\prime}\otimes\chi)}}L(s,\pi\times(\pi^{\prime}\otimes\chi))$

and its derivative in the region $\operatorname{Re}(s)\geq 0$.

######  Lemma 7.4.

For all $\sigma\geq 0$, $t\in\mathbb{R}$, $j\in\\{0,1\\}$, and
$\varepsilon>0$,

$\mathcal{L}^{(j)}(\sigma+it,\pi\times(\pi^{\prime}\otimes\chi))\ll_{\pi,\pi^{\prime},\varepsilon}\big{(}q_{\chi}(|t|+3)\big{)}^{mm^{\prime}\max(1-\sigma,0)/2+\varepsilon}.$

Moreover, if $r_{\pi\times(\pi^{\prime}\otimes\chi)}=0$ or
$|\sigma+it-1|>1/4$, then

(7.9)
$L^{(j)}(\sigma+it,\pi\times(\pi^{\prime}\otimes\chi))\ll_{\pi,\pi^{\prime},\varepsilon}\big{(}q_{\chi}(|t|+3)\big{)}^{mm^{\prime}\max(1-\sigma,0)/2+\varepsilon}.$

###### Proof.

See [19, Lemma 3.2]. ∎

##  8\. Symmetric Power $L$-functions

###  8.1. Automorphy of symmetric powers

Let $f\in S_{k}^{\mathrm{new}}(\Gamma_{0}(N))$ be a newform as in Section 1.
For each $m\in\mathbb{N}$, there exists a local factor at the primes $p\nmid
N$ of the form

(8.1)
$L_{p}(s,\operatorname{Sym}^{m}f):=\prod_{j=0}^{m}(1-\alpha_{j,\operatorname{Sym}^{m}f}(p)p^{-s})^{-1},\qquad\alpha_{j,\operatorname{Sym}^{m}f}(p):=e^{i(m-2j)\theta_{f}(p)}.$

For each prime $p\mid N$, there exists a local factor
$L_{p}(s,\operatorname{Sym}^{m}f)$ such that the $m$-th symmetric power
$L$-function of $f$ is given by

$L(s,\operatorname{Sym}^{m}f):=\prod_{p\mid
N}L_{p}(s,\operatorname{Sym}^{m}f)\prod_{p\nmid
N}\prod_{j=0}^{m}(1-e^{i(m-2j)\theta_{f}(p)}p^{-s})^{-1}=\prod_{p}\prod_{j=0}^{m}(1-\alpha_{j,\operatorname{Sym}^{m}f}(p)p^{-s})^{-1},$

which converges absolutely for $\operatorname{Re}(s)>1$.

######  Theorem 8.1 (Newton and Thorne, [30, 31]).

Let $f$ be a non-CM holomorphic newform with trivial nebentypus. Then, there
exists $\pi\in\mathfrak{F}_{m+1}$ such that
$L(s,\operatorname{Sym}^{m}f)=L(s,\pi)$.

For non-CM holomorphic newforms $f_{1}$ and $f_{2}$, recall that the notation
$f_{1}\not\sim f_{2}$ means that if $\psi$ is a primitive Dirichlet character,
then $f_{1}\neq f_{2}\otimes\psi$. Since the nebentypus characters of $f_{1}$
and $f_{2}$ are trivial, it follows from Theorem 8.1 that there exist self-
dual representations $\pi\in\mathfrak{F}_{m_{1}+1}$ and
$\pi^{\prime}\in\mathfrak{F}_{m_{2}+1}$ such that

$L(s,\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi))=L(s,\pi\times(\pi^{\prime}\otimes\chi)).$

In order to determine whether
$L(s,\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi))$
has a pole at $s=1$ or not, we have the following proposition which follows
from the work of Rajan [38, Corollary 5.1].

######  Proposition 8.2.

If $f_{1}\not\sim f_{2}$ and $(m_{1},m_{2})\neq(0,0)$, then
$L(s,\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi))$
is entire.

###### Proof.

By Theorem 8.1, let $\pi,\pi^{\prime}$ be the cuspidal automorphic
representations associated to $\operatorname{Sym}^{m_{1}}f_{1}$ and
$\operatorname{Sym}^{m_{2}}f_{2}$ respectively. Notice that $\pi$ and
$\pi^{\prime}$ are self-dual because our newforms have trivial nebentypus.
Then, by Theorem 7.1, it suffices to show that
$\pi\ncong\pi^{\prime}\otimes\chi$. If $m_{1}\neq m_{2}$, then
$\pi\ncong\pi^{\prime}\otimes\chi$ because $\pi$ has degree $m_{1}+1$ and
$\pi^{\prime}\otimes\chi$ has degree $m_{2}+1$. Therefore, we may assume
without loss of generality that $m_{1}=m_{2}=\ell>0$.

Rajan [38, Corollary 5.1] proved that if

(8.2) $\limsup_{x\to\infty}\frac{\\#\\{p\leq
x:U_{\ell}(\cos\theta_{1}(p))=U_{\ell}(\cos\theta_{2}(p))\\}}{\pi(x)}>0,$

then there exists a Dirichlet character $\psi$ such that
$f_{1}=f_{2}\otimes\psi$. Thus, assume for the sake of contradiction that
$\pi=\pi^{\prime}\otimes\chi$. This implies that for all but finitely many
primes $p$, one has

$U_{\ell}(\cos\theta_{1}(p))=U_{\ell}(\cos\theta_{2}(p))\chi(p).$

Therefore, (8.2) holds because the (natural) density of primes $p$ such that
$\chi(p)=1$ equals $1/\varphi(q_{\chi})$. This implies that $f_{1}$ and
$f_{2}$ are related by twist, which is a contradiction. The proposition
follows. ∎

The following lemma will be useful later.

######  Lemma 8.3.

Let $f$ be a holomorphic newform with level $N$ and trivial nebentypus. For
all $m_{1},m_{2}\in\mathbb{N}$, one has

$L(s,\operatorname{Sym}^{m_{1}}f\times\operatorname{Sym}^{m_{2}}f)=\prod_{j=0}^{\min(m_{1},m_{2})}L(s,\operatorname{Sym}^{|m_{1}-m_{2}|+2j}f).$

###### Proof.

This follows from the Clebsch–Gordan identities for ${\rm SU}(2)$. ∎

###  8.2. Estimates on Dirichlet Coefficients

Note that in applications (7.5) holds, which allows us to assume for our
purposes of bounding coefficients that $\chi$ is trivial. We say that an
$L$-function satisfies the Generalized Ramanujan Conjecture (GRC) if its
Satake parameters are bounded in modulus by $1$, and its Langlands parameters
have nonnegative real parts. We have the following two lemmata.

######  Lemma 8.4.

Let $f$ be a holomorphic newform with level $N$ and trivial nebentypus. Then,
for every $m\in\mathbb{N}\cup\\{0\\}$, $L(s,\operatorname{Sym}^{m}f)$
satisfies GRC.

###### Proof.

For $m=0$, $L(s,\operatorname{Sym}^{0}f)=\zeta(s)$ satisfies GRC. Therefore,
let us assume $m\in\mathbb{N}$. If $p$ is a prime not dividing $N$, then the
Satake parameters of $L_{p}(s,\operatorname{Sym}^{m}f)$ are bounded in modulus
by $1$ by considering (8.1). To show that the other primes $p\mid N$ have
local factors whose parameters are bounded in modulus by $1$, as well as
showing that the Langlands parameters have nonnegative real parts, we proceed
as in [1, Proof of Corollary 7.1.15]. ∎

######  Lemma 8.5.

Let $f_{1}$ and $f_{2}$ be non-CM holomorphic newforms. For all
$m_{1},m_{2}\in\mathbb{N}\cup\\{0\\}$ and $n\in\mathbb{N}$,

(8.3)
$\displaystyle|\Lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times\operatorname{Sym}^{m_{2}}f_{2}}(n)|$
$\displaystyle\leq(m_{1}+1)(m_{2}+1)\Lambda(n),$ (8.4)
$\displaystyle|\lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times\operatorname{Sym}^{m_{2}}f_{2}}(n)|$
$\displaystyle\leq d_{(m_{1}+1)(m_{2}+1)}(n),$ (8.5)
$\displaystyle|\mu_{\operatorname{Sym}^{m_{1}}f_{1}\times\operatorname{Sym}^{m_{2}}f_{2}}(n)|$
$\displaystyle\ll_{m_{1},m_{2}}d_{2}(n)^{(m_{1}+1)(m_{2}+1)}.$

###### Proof.

We apply (7.7) with $\pi=\operatorname{Sym}^{m_{1}}f_{1}$,
$\pi^{\prime}=\operatorname{Sym}^{m_{2}}f_{2}$ in order to obtain

$|\Lambda_{\pi\times\pi^{\prime}}(n)|\leq\sqrt{\Lambda_{\pi\times\pi}(n)\Lambda_{\pi^{\prime}\times\pi^{\prime}}(n)}.$

When $n$ is not a prime power, $\Lambda_{\pi\times\pi^{\prime}}(n)=0$. When
$n=p^{\ell}$ is a prime power, Lemma 8.3 implies that

$\Lambda_{\pi\times\pi}(p^{\ell})=\sum_{j=0}^{m_{1}}\Lambda_{\operatorname{Sym}^{2j}f_{1}}(p^{\ell})=\sum_{j=0}^{m_{1}}\sum_{r=0}^{2j}\alpha_{r,\operatorname{Sym}^{2j}f_{1}}(p)^{\ell}\log
p,$

and similarly for $\operatorname{Sym}^{m_{2}}f_{2}$. The bound (8.3) now
follows from Lemma 8.4.

For the proof of (8.4), one relates $\lambda_{\pi\times\pi^{\prime}}(n)$ to
$\Lambda_{\pi\times\pi^{\prime}}(n)$ by formally relating the Dirichlet series
$L(s,\pi\times\pi^{\prime})$ and
$-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime})$. We reduce to the case of
prime powers, since $\lambda_{\pi\times\pi^{\prime}}(n)$ is a multiplicative
function. For $n=p^{\ell}$, applying formal integration and exponentiation to
$-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime})$ yields the relation between
Dirichlet coefficients

(8.6)
$\lambda_{\pi\times\pi^{\prime}}(p^{\ell})=\sum_{r=1}^{\ell}\frac{1}{r!}\sum_{j_{1}+\cdots+j_{r}=\ell}\frac{\Lambda_{\pi\times\pi^{\prime}}(p^{j_{1}})\cdots\Lambda_{\pi\times\pi^{\prime}}(p^{j_{r}})}{j_{1}\cdots
j_{r}(\log p)^{r}},\qquad j_{i}\in\mathbb{N}.$

Applying (8.3), we obtain (8.4) since

$|\lambda_{\pi\times\pi^{\prime}}(p^{\ell})|\leq\sum_{r=1}^{\ell}\frac{[(m_{1}+1)(m_{2}+1)]^{r}}{r!}\sum_{j_{1}+\cdots+j_{r}=\ell}\frac{1}{j_{1}\cdots
j_{r}}=d_{(m_{1}+1)(m_{2}+1)}(p^{\ell}).$

Lastly, we prove (8.5). For all prime powers $p^{\ell}$, we have that

$\mu_{\pi\times\pi^{\prime}}(p^{\ell})=(-1)^{\ell}\sum_{j_{1}<\cdots<j_{\ell}}\alpha_{j_{1},\pi\times\pi^{\prime}}(p)\cdots\alpha_{j_{\ell},\pi\times\pi^{\prime}}(p),\qquad
1\leq j_{i}\leq(m_{1}+1)(m_{2}+1).$

In particular, $\mu_{\pi\times\pi^{\prime}}(p^{\ell})=0$ for
$\ell>(m_{1}+1)(m_{2}+1)$. Also, if $p\nmid N_{1}N_{2}$, then

$|\mu(p^{\ell})|\leq\binom{(m_{1}+1)(m_{2}+1)}{\ell}\leq(\ell+1)^{(m_{1}+1)(m_{2}+1)}=d_{2}(p^{\ell})^{(m_{1}+1)(m_{2}+1)}.$

Thus, because there are only finitely many $p\mid N_{1}N_{2}$, we deduce by
the multiplicativity of $\mu_{\pi\times\pi^{\prime}}(n)$ and $d_{2}(n)$ that
(8.5) holds. ∎

We have the following lemma due to Norton [32] that allows us to obtain
summatory averages of products of divisor functions.

######  Lemma 8.6.

Let $x\geq 2$ and $j_{1},\ldots,j_{r}\in\mathbb{N}$. Then, there exists a
constant $c_{8}=c_{8}(j_{1},\ldots,j_{r})>0$ such that

$\sum_{n\leq x}d_{j_{1}}(n)\cdots d_{j_{r}}(n)\ll_{j_{1},\ldots,j_{r}}x(\log
x)^{c_{8}}$

###### Proof.

See [32, Theorem 1.7]. ∎

###  8.3. Sums of Dirichlet Coefficients

We now address sums of the form

$\sum_{n\leq
x}\lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}^{\prime}\otimes\chi)}(n).$

We have the following lemma that is sufficient for the regime of conductors
used in Vaughan’s method in Section 9.

######  Lemma 8.7.

Let $\chi$ be a primitive Dirichlet character modulo $q_{\chi}$ with
$(q_{\chi},N_{1}N_{2})=1$. Let $\theta\in(0,\frac{2}{(m_{1}+1)(m_{2}+1)})$ and
$c(\theta)=\frac{2-(m_{1}+1)(m_{2}+1)\theta}{(m_{1}+1)(m_{2}+1)+1}$. We have
that if $q_{\chi}\leq x^{\theta}$, then for all $\varepsilon>0$,

$\sum_{n\leq
x}\lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}(n)\ll_{m_{1},m_{2},\theta,\varepsilon}x^{1-c(\theta)+\varepsilon}$

holds uniformly over all such $\chi$.

###### Proof.

By 7.5, 8.4, and the fact that $d_{r}(n)\ll_{r,\varepsilon}n^{\varepsilon}$,
one has that

$|\lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}(n)|\leq
d_{(m_{1}+1)(m_{2}+1)}(n)\ll_{m_{1},m_{2},\varepsilon}n^{\varepsilon}.$

Therefore,
$L(s,\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi))$
satisfies the hypotheses of [13, Proposition 1.1]. Therefore, we have for
$x\geq
q_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}^{1/2}$
that

$\sum_{n\leq
x}\lambda_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}\ll
q_{\chi}^{\frac{(m_{1}+1)(m_{2}+1)}{(m_{1}+1)(m_{2}+1)+1}}x^{\frac{(m_{1}+1)(m_{2}+1)-1}{(m_{1}+1)(m_{2}+1)+1}+\varepsilon}\leq
x^{1-c(\theta)+\varepsilon},$

where the implied constant only depends on
$f_{1},f_{2},m_{1},m_{2},\varepsilon$, and the Langlands parameters of
$L(s,\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi))$.
One may observe that the Langlands parameters only depend on the weights
$k_{1},k_{2}$ of $f_{1},f_{2}$ respectively and $\chi(-1)=\pm 1$. Therefore,
the implied constant only depends on $f_{1},f_{2},m_{1},m_{2},\varepsilon$.

In order to handle $x\leq
q_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}^{1/2}$,
we observe by (7.2) that

$q_{\operatorname{Sym}^{m_{1}}f_{1}\times(\operatorname{Sym}^{m_{2}}f_{2}\otimes\chi)}^{1/2}\ll_{m_{1},m_{2}}q_{\chi}^{(m_{1}+1)(m_{2}+1)/2}\leq
x^{\theta(m_{1}+1)(m_{2}+1)/2}.$

Since $\theta(m_{1}+1)(m_{2}+1)/2<1$, this range of $x$ is bounded in terms of
$f_{1},f_{2},m_{1},m_{2},\theta$, and so by allowing the implied constant in
our lemma to depend on $\theta$, we are done. ∎

##  9\. Bombieri–Vinogradov

Let $\pi=\operatorname{Sym}^{m_{1}}f_{1}$ and
$\pi^{\prime}=\operatorname{Sym}^{m_{2}}f_{2}$ where $(m_{1},m_{2})\neq(0,0)$.
For each $q\geq 1$ and $a\in(\mathbb{Z}/q\mathbb{Z})^{\times}$, we define

$\psi_{\pi\times\pi^{\prime}}(x;q,a):=\sum_{\begin{subarray}{c}n\leq x\\\
n\equiv a\ (\mathrm{mod}\
q)\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(n).$

Let $m=m_{1}+1$ and $m^{\prime}=m_{2}+1$. In order to prove Theorem 3.3, our
goal is to prove the following theorem.

######  Theorem 9.1.

For any level of distribution $\theta\in(0,\frac{2}{2mm^{\prime}+1})$, one has
for any $A>0$ that

(9.1) $\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\max_{\begin{subarray}{c}y\leq
x\\\
(a,q)=1\end{subarray}}|\psi_{\pi\times\pi^{\prime}}(y;q,a)|\ll\frac{x}{(\log
x)^{A}},$

where the implied constant only depends on $\pi,\pi^{\prime},\theta,A$.

First, we handle the contribution from moduli $q$ no larger than a fixed power
of $\log x$. The following result is proven by the recent work of Harcos and
Thorner [19].

######  Theorem 9.2.

If $A>0$ is fixed, then

$\sum_{\begin{subarray}{c}q\leq(\log x)^{A}\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\max_{\begin{subarray}{c}y\leq
x\\\
(a,q)=1\end{subarray}}|\psi_{\pi\times\pi^{\prime}}(y;q,a)|\ll\frac{x}{(\log
x)^{A}},$

where the implied constant only depends on $\pi,\pi^{\prime},\theta,A$.

###### Proof.

Per Proposition 8.2, if $\chi\ (\mathrm{mod}\ q_{\chi})$ is primitive, and
$\gcd(q_{\chi},q_{\pi}q_{\pi^{\prime}})=1$, then
$L(s,\pi\times(\pi^{\prime}\otimes\chi))$ is entire. Therefore, the result
follows from [19, Theorem 2.1]. ∎

Throughout, $U,V\geq 1$ are parameters that we will take to be small powers of
$x$, in particular we may assume $\log U\ll\log x$ and $\log V\ll\log x$. For
$\theta\in(0,\frac{2}{mm^{\prime}})$, define $c(\theta):=\frac{2-\theta
mm^{\prime}}{mm^{\prime}+1}$. We begin by proving the following theorem for

$\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(x):=\sum_{n\leq
x}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n).$

######  Theorem 9.3.

Let $\theta\in(0,\frac{2}{mm^{\prime}})$. Then, for all $x\geq 1$,
$Q\in[1,x^{\theta}]$, there exists $c_{9}=c_{9}(m,m^{\prime})>0$ such that for
any $\varepsilon\in(0,\frac{1}{2})$,

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)|\\\
\ll_{\pi,\pi^{\prime},\theta,\varepsilon}(Q^{2}x^{1/2}+x+Qx^{1-\varepsilon/2}+Qx^{1/2+\varepsilon}+Q^{2}x^{1-c(\theta)+\varepsilon})(\log
x)^{c_{9}}.$

In order to prove Theorem 9.3, we employ an analogue of Vaughan’s identity,
namely

$\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)=\mathfrak{a}_{1}(n)+\mathfrak{a}_{2}(n)+\mathfrak{a}_{3}(n)+\mathfrak{a}_{4}(n),$

where

$\displaystyle\mathfrak{a}_{1}(n)$
$\displaystyle:=\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)\mathbbm{1}_{n\leq
U},$ $\displaystyle\mathfrak{a}_{2}(n)$
$\displaystyle:=-\sum_{\begin{subarray}{c}j\ell r=n\\\ j\leq U\\\ \ell\leq
V\end{subarray}}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(j)\mu_{\pi\times(\pi^{\prime}\otimes\chi)}(\ell)\lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(r),$
$\displaystyle\mathfrak{a}_{3}(n)$
$\displaystyle:=\sum_{\begin{subarray}{c}h\ell=n\\\ \ell\leq
V\end{subarray}}\mu_{\pi\times(\pi^{\prime}\otimes\chi)}(\ell)\lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(h)\log(h),$
$\displaystyle\mathfrak{a}_{4}(n)$
$\displaystyle:=-\sum_{\begin{subarray}{c}jk=n\\\ k>1\\\
j>U\end{subarray}}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(j)\Big{(}\sum_{\begin{subarray}{c}\ell
r=k\\\ \ell\leq
V\end{subarray}}\lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(r)\mu_{\pi\times(\pi^{\prime}\otimes\chi)}(\ell)\Big{)}.$

Therefore, it suffices to estimate the average over primitive characters of
the following sums.

$\displaystyle S_{j}$ $\displaystyle:=\sum_{n\leq y}\mathfrak{a}_{j}(n),$
$\displaystyle\Sigma_{j}$ $\displaystyle:=\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{j}|,\qquad j\in\\{1,2,3,4\\}.$

Using (8.3), we estimate that

(9.2)
$S_{1}=\sum_{n\leq\min\\{y,U\\}}\Lambda_{\pi\times(\pi^{\prime}\otimes\chi)}(n)\ll_{\pi,\pi^{\prime}}U,\qquad\Sigma_{1}\ll_{\pi,\pi^{\prime}}Q^{2}U.$

To treat $S_{2}$, by definition

$S_{2}=-\sum_{n\leq y}\sum_{\begin{subarray}{c}j\ell r=n\\\ j\leq U\\\
\ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\lambda_{\pi\times\pi^{\prime}}(r)\chi(n).$

Putting $t=j\ell$ and swapping order of summation, we have that

$S_{2}=-\sum_{t\leq UV}\Big{(}\sum_{\begin{subarray}{c}j\ell=t\\\ j\leq U\\\
\ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\sum_{r\leq
y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(rt).$

We split the sum over $t$ into the ranges $t\leq U$ and $U<t\leq UV$. We
define

$\displaystyle S_{2}^{\prime}$ $\displaystyle=-\sum_{t\leq
U}\Big{(}\sum_{\begin{subarray}{c}j\ell=t\\\ j\leq U\\\ \ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\sum_{r\leq
y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(rt),$ $\displaystyle
S_{2}^{\prime\prime}$ $\displaystyle=-\sum_{U<t\leq
UV}\Big{(}\sum_{\begin{subarray}{c}j\ell=t\\\ j\leq U\\\ \ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\sum_{r\leq
y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(rt),$
$\displaystyle\Sigma_{2}^{\prime}$
$\displaystyle:=\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{2}^{\prime}|,$ $\displaystyle\Sigma_{2}^{\prime\prime}$
$\displaystyle:=\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{2}^{\prime\prime}|.$

Notice that $S_{2}=S_{2}^{\prime}+S_{2}^{\prime\prime}$ and
$\Sigma_{2}\leq\Sigma_{2}^{\prime}+\Sigma_{2}^{\prime\prime}$.

###  9.1. Large Sieve Terms

We handle the terms $S_{2}^{\prime\prime}$ and $S_{4}$ using the following
bilinear large sieve inequality derived in Davenport [10].

######  Proposition 9.4.

Let $\\{a_{j}\\},\\{b_{k}\\}$ be sequences of complex numbers. If $Q,J,K\geq
1$, then

$\sum_{q\leq Q}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{u\leq
JK}\Big{|}\underset{jk\leq u}{\sum_{1\leq j\leq J}\sum_{1\leq k\leq
K}}a_{j}b_{k}\chi(jk)\Big{|}\\\
\ll(Q^{2}+J)^{1/2}(Q^{2}+K)^{1/2}\Big{(}\sum_{1\leq j\leq
J}|a_{j}|^{2}\Big{)}^{1/2}\Big{(}\sum_{1\leq k\leq
K}|b_{k}|^{2}\Big{)}^{1/2}\log(2JK),$

where the implied constant is absolute and effective.

###### Proof.

See [10, Chapter 28]. ∎

To treat $S_{4}$, we use the fact that if $q_{\chi}$ is coprime to
$q_{\pi}q_{\pi^{\prime}}$, we have (7.5). Therefore, we may write

$S_{4}=\sum_{U<j\leq y/V}\Lambda_{\pi\times\pi^{\prime}}(j)\sum_{V<k\leq
y/j}\Big{(}\sum_{\begin{subarray}{c}\ell r=k\\\ \ell\leq
V\end{subarray}}\lambda_{\pi\times\pi^{\prime}}(r)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\chi(jk).$

We now consider a dyadic partition of the range $j$. For $1\leq J\leq x$, Let

$S_{4}^{(J)}:=\sum_{\begin{subarray}{c}U<j\leq y/V\\\ J<j\leq
2J\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\sum_{V<k\leq
y/j}\Big{(}\sum_{\begin{subarray}{c}\ell r=k\\\ \ell\leq
V\end{subarray}}\lambda_{\pi\times\pi^{\prime}}(r)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\chi(jk).$

Applying Proposition 9.4 with $K=x/J$, we have that

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{4}^{(J)}|\ll(Q^{2}+J)^{1/2}\Big{(}Q^{2}+\frac{x}{J}\Big{)}^{1/2}\Big{(}\sum_{J\leq
m\leq 2J}|\Lambda_{\pi\times\pi^{\prime}}(j)|^{2}\Big{)}^{1/2}\\\
\times\Big{(}\sum_{k\leq x/J}\Big{|}\sum_{\begin{subarray}{c}\ell r=k\\\
\ell\leq
V\end{subarray}}\lambda_{\pi\times\pi^{\prime}}(r)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{|}^{2}\Big{)}^{1/2}\log(x).$

Using Lemmas 8.5 and 8.6, there exists $c_{10}=c_{10}(m,m^{\prime})>0$ such
that

$\displaystyle\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{4}^{(J)}|$
$\displaystyle\ll_{\pi,\pi^{\prime}}(Q^{2}+J)^{1/2}\Big{(}Q^{2}+\frac{x}{J}\Big{)}^{1/2}(J\log
J)^{1/2}x^{1/2}(\log x)^{c_{10}}$
$\displaystyle\ll(Q^{2}x^{1/2}+QxJ^{-1/2}+Qx^{1/2}J^{1/2}+x)(\log
x)^{c_{10}}.$

Summing dyadically over $[J,2J]$ so that the range $[U,x/V]$ is covered, this
yields

(9.3)
$\Sigma_{4}\ll_{\pi,\pi^{\prime}}(Q^{2}x^{1/2}+QxU^{-1/2}+QxV^{-1/2}+x)(\log
x)^{c_{10}}.$

The sum $S_{2}^{\prime\prime}$ is handled similarly to $S_{4}$. We dyadically
decompose $S_{2}^{\prime\prime}$ into the sums

$S_{2}^{(T)}:=-\sum_{T\leq t\leq 2T}\Big{(}\sum_{\begin{subarray}{c}j\ell=t\\\
j\leq U\\\ \ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{)}\sum_{r\leq
y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(rt).$

Applying Proposition 9.4 with $J=T$ and $K=x/T$ yields

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|S_{2}^{(T)}|\ll(T+Q^{2})^{1/2}\Big{(}\frac{x}{T}+Q^{2}\Big{)}^{1/2}\Big{(}\sum_{T\leq
t\leq 2T}\Big{|}\sum_{\begin{subarray}{c}j\ell=t\\\ j\leq U\\\ \ell\leq
V\end{subarray}}\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)\Big{|}^{2}\Big{)}^{1/2}\\\
\times\Big{(}\sum_{r\leq
x/T}\Big{|}\lambda_{\pi\times\pi^{\prime}}(r)\Big{|}^{2}\Big{)}^{1/2}\log(x).$

Using Lemmas 8.5 and 8.6, there exists $c_{11}=c_{11}(m,m^{\prime})>0$ such
that

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}\Big{|}S_{2}^{(T)}\Big{|}\ll_{\pi,\pi^{\prime}}(Q^{2}x^{1/2}+QxT^{-1/2}+Qx^{1/2}T^{1/2}+x)(\log
x)^{c_{11}},$

Summing dyadically over $[T,2T]$ so that the range $[U,UV]$ is covered, we
obtain

(9.4)
$\Sigma_{2}^{\prime\prime}\ll_{\pi,\pi^{\prime}}(Q^{2}x^{1/2}+QxU^{-1/2}+Qx^{1/2}U^{1/2}V^{1/2}+x)(\log
x)^{c_{11}}.$

###  9.2. Terms with sums of Dirichlet coefficients

We handle the terms $S_{2}^{\prime}$ and $S_{3}$ using Lemma 8.7.

To treat $S_{2}^{\prime}$, we first have from (7.5), Lemma 8.5, and factoring
out $|\chi(t)|\leq 1$,

$\displaystyle|S_{2}^{\prime}|$ $\displaystyle\leq\sum_{t\leq
U}\Big{(}\sum_{\begin{subarray}{c}j\ell=t\\\ \ell\leq
V\end{subarray}}|\Lambda_{\pi\times\pi^{\prime}}(j)\mu_{\pi\times\pi^{\prime}}(\ell)|\Big{)}\Big{|}\sum_{r\leq
y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(rt)\Big{|}$ (9.5)
$\displaystyle\ll_{\pi,\pi^{\prime}}(\log
U)\cdot\Big{(}\sum_{\ell\leq\min(U,V)}d_{2}(\ell)^{mm^{\prime}}\Big{)}\cdot\sum_{t\leq
U}\Big{|}\sum_{r\leq y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(r)\Big{|}.$

Applying Lemmas 8.5 and 8.6, there exists $c_{12}=c_{12}(m,m^{\prime})>0$ such
that

$\ll_{\pi,\pi^{\prime}}\min(U,V)(\log x)^{c_{12}}\cdot\sum_{t\leq
U}\Big{|}\sum_{r\leq y/t}\lambda_{\pi\times\pi^{\prime}}(r)\chi(r)\Big{|}.$

Thus, by Lemma 8.7,

$|S_{2}^{\prime}|\ll_{\pi,\pi^{\prime},\theta,\varepsilon}\min(U,V)(\log
x)^{c_{12}}\sum_{t\leq
U}(y/t)^{1-c(\theta)+\varepsilon}\ll_{m,m^{\prime},\theta,\varepsilon}\min(U,V)x^{1-c(\theta)+\varepsilon}U^{c(\theta)}(\log
x)^{c_{12}}.$

To treat $S_{3}$, we exchange the order of summation to obtain

$S_{3}=\sum_{\ell\leq V}\mu_{\pi\times\pi^{\prime}}(\ell)\sum_{h\leq
y/\ell}\lambda_{\pi\times\pi^{\prime}}(h)\log(h)\chi(\ell h).$

Summing by parts and factoring out $\chi(\ell)$, we obtain

$\sum_{h\leq y/\ell}\lambda_{\pi\times\pi^{\prime}}(h)\log(h)\chi(\ell
h)=\log(y/\ell)\chi(\ell)\sum_{h\leq
y/\ell}\lambda_{\pi\times\pi^{\prime}}(h)\chi(h)-\chi(\ell)\int_{1}^{y/\ell}\Big{(}\sum_{h\leq
u}\lambda_{\pi\times\pi^{\prime}}(h)\chi(h)\Big{)}\,\frac{du}{u}.$

Applying Lemma 8.7 to the sums on the right, we see that

$\displaystyle\sum_{h\leq
y/\ell}\lambda_{\pi\times\pi^{\prime}}(h)\log(h)\chi(\ell h)$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}[\log(y/\ell)](y/\ell)^{1-c(\theta)+\varepsilon}+\int_{1}^{y/\ell}u^{1-c(\theta)+\varepsilon}\,\frac{du}{u}$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}(\log
y/\ell)(y/\ell)^{1-c(\theta)+\varepsilon}.$

So, we have

$|S_{3}|\ll_{\pi,\pi^{\prime},\theta,\varepsilon}(\log x)\sum_{\ell\leq
V}|\mu_{\pi\times\pi^{\prime}}(\ell)|(y/\ell)^{1-c(\theta)+\varepsilon}.$

Applying Lemmas 8.5 and 8.6 and summing by parts, there exists
$c_{13}=c_{13}(m,m^{\prime})>0$ such that

$\displaystyle|S_{3}|$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}x^{1-c^{\prime}}(\log
x)\sum_{\ell\leq V}d_{2}(\ell)^{mm^{\prime}}/\ell^{1-c(\theta)+\varepsilon}$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}x^{1-c(\theta)+\varepsilon}V^{c(\theta)}(\log
x)^{c_{13}}.$

Our bounds on $S_{2}^{\prime}$ and $S_{3}$ yield the following upper bounds:

(9.6) $\displaystyle\Sigma_{2}^{\prime}$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}Q^{2}x^{1-c(\theta)+\varepsilon}U^{1+c(\theta)}(\log
x)^{c_{12}};$ (9.7) $\displaystyle\Sigma_{3}$
$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}Q^{2}x^{1-c(\theta)+\varepsilon}V^{c(\theta)}(\log
x)^{c_{13}}.$

###  9.3. Conclusion of Bombieri–Vinogradov

###### Proof of Theorem 9.3.

Piecing together 9.2–9.4, 9.6, and 9.7, there exists
$c_{9}=c_{9}(m,m^{\prime})>0$ such that

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)|\ll_{\pi,\pi^{\prime},\theta,\varepsilon}\\\
(Q^{2}x^{1/2}+x+QxU^{-1/2}+QxV^{-1/2}+U^{1/2}V^{1/2}Qx^{1/2}+Q^{2}x^{1-c(\theta)+\varepsilon}(U^{1+c(\theta)}+V^{c(\theta)}))(\log
x)^{c_{9}}.$

Setting $U=V=x^{\varepsilon}$, where $\varepsilon\in(0,\frac{1}{2})$, this
yields Theorem 9.3 by rescaling $\varepsilon$. ∎

###### Proof of Theorem 9.1.

Let $Q=x^{\theta}$. By character orthogonality, we may write

$\psi_{\pi\times\pi^{\prime}}(y,\chi):=\sum_{n\leq
y}\Lambda_{\pi\times\pi^{\prime}}(n)\chi(n),\qquad\psi_{\pi\times\pi^{\prime}}(y;q,a)=\frac{1}{\varphi(q)}\sum_{\chi\
(\mathrm{mod}\ q)}\overline{\chi}(a)\psi_{\pi\times\pi^{\prime}}(y,\chi).$

Note that the conductor $q$ is coprime to $q_{\pi}q_{\pi^{\prime}}$ and $\chi$
is primitive, so we have that
$\psi_{\pi\times\pi^{\prime}}(y,\chi)=\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)$.
Using the above expression for $\psi_{\pi\times\pi^{\prime}}(y;q,a)$, we have
that the left hand side of (9.1) is

$\ll\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{1}{\varphi(q)}\sum_{\chi\
(\mathrm{mod}\ q)}\max_{y\leq x}|\psi_{\pi\times\pi^{\prime}}(y,\chi)|.$

We are able to convert between $\chi$ and the primitive character that induces
it, call it $\chi_{1}$, at a negligible cost. Indeed,

$|\psi_{\pi\times\pi^{\prime}}(y,\chi)-\psi_{\pi\times\pi^{\prime}}(y,\chi_{1})|\leq\sum_{\begin{subarray}{c}p^{\ell}\leq
y\\\ p\mid
q_{\chi}q_{\pi}q_{\pi^{\prime}}\end{subarray}}|\Lambda_{\pi\times\pi^{\prime}}(p^{\ell})|\ll_{\pi,\pi^{\prime}}\sum_{p\mid
q_{\chi}q_{\pi}q_{\pi^{\prime}}}\Big{\lfloor}\frac{\log y}{\log
p}\Big{\rfloor}\log(p)\ll_{\pi,\pi^{\prime}}(\log q_{\chi}y)^{2}.$

Originally, each character $\chi\ (\mathrm{mod}\ q)$ is assigned a weighting
$1/\varphi(q)$. When reducing to primitive characters, each primitive
character $\chi\ (\mathrm{mod}\ q)$ is assigned a weighting $\sum_{k\leq
Q/q}\frac{1}{\varphi(kq)}$ that is $\ll\frac{\log Q}{\varphi(q)}$, since we
may observe that $\varphi(kq)\geq\varphi(k)\varphi(q)$ and then apply an
estimate of Mertens. Thus, the left hand side of $\eqref{eqn:BV Main Thm}$ is

$\ll_{\pi,\pi^{\prime}}Q(\log Qx)^{2}+(\log Q)\sum_{\begin{subarray}{c}q\leq
Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{1}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)|.$

The term $Q(\log Qx)^{2}\ll_{\theta}x^{\theta}(\log x)^{2}$ is negligible
since $\theta<1$, and the factor of $\log Q$ in front of our primitive
character sum is also negligible, since it is $\ll_{\theta}\log x$ and we are
saving arbitrary log powers. It therefore suffices to show that the sum over
moduli is $\ll_{\pi,\pi^{\prime},\theta,A}x(\log x)^{-A}$. By a dyadic
partition, we have that

$\sum_{\begin{subarray}{c}q\leq Q\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\frac{1}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\
(\mathrm{mod}\ q)\\\ \textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)|\ll\frac{(\log
Q)}{R}\max_{R\leq Q}\sum_{R\leq q\leq
2R}\frac{q}{\varphi(q)}\sum_{\begin{subarray}{c}\chi\ (\mathrm{mod}\ q)\\\
\textup{$\chi$ primitive}\end{subarray}}\max_{y\leq
x}|\psi_{\pi\times(\pi^{\prime}\otimes\chi)}(y)|.$

If the maximum occurs at $R\leq(\log x)^{A^{\prime}}$, where
$A^{\prime}=A^{\prime}(m,m^{\prime})>0$ will be chosen later, then we apply
Theorem 9.2 in order to conclude the proof.

We now consider the case where the maximum occurs at $R\geq(\log
x)^{A^{\prime}}$. By applying Theorem 9.3, we have that the left hand side of
(9.1) is

$\displaystyle\ll_{\pi,\pi^{\prime},\theta,\varepsilon}\max_{(\log
x)^{A^{\prime}}\leq R\leq
Q}(x^{1/2}R+xR^{-1}+x^{1-\varepsilon/2}+x^{1-c(\theta)+\varepsilon}R)(\log
x)^{c_{9}}$ (9.8) $\displaystyle\ll_{\varepsilon,A^{\prime}}x(\log
x)^{c_{9}-A^{\prime}}+Qx^{1-c(\theta)+\varepsilon}(\log x)^{c_{9}}.$

It therefore suffices to take $A^{\prime}-c_{9}\geq A$ and
$x^{1+\theta-c(\theta)+\varepsilon}=o(x)$. The latter condition says that

$\theta<\frac{2}{2mm^{\prime}+1}-\varepsilon\Big{(}\frac{mm^{\prime}+1}{2mm^{\prime}+1}\Big{)}.$

We now take $\varepsilon=\varepsilon(m,m^{\prime},\theta)$ sufficiently small
so that the above inequality holds for any choice of
$\theta\in(0,\frac{2}{2mm^{\prime}+1})$. ∎

We may now pass from Theorem 9.1, which is counting prime powers with von
Mangoldt weights, to counting primes. This is then used to prove Theorem 3.3.

######  Theorem 9.5.

Let $f_{1}\not\sim f_{2}$ be non-CM holomorphic newforms with levels
$N_{1},N_{2}$ respectively. Let $m_{1},m_{2}\in\mathbb{N}\cup\\{0\\}$ such
that $(m_{1},m_{2})\neq(0,0)$. For any
$\theta\in\Big{(}0,\frac{2}{2(m_{1}+1)(m_{2}+1)+1}\Big{)}$ and $A>0$,

$\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
(q,N_{1}N_{2})=1\end{subarray}}\max_{(a,q)=1}\Big{|}\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv
a(q)\end{subarray}}U_{m_{1}}(\cos\theta_{1}(p))U_{m_{2}}(\cos\theta_{2}(p))\Big{|}\ll\frac{x}{(\log
x)^{A}},$

where the implied constant only depends on $f_{1},f_{2},m_{1},m_{2},\theta,A$.

###### Proof.

Let $\pi=\operatorname{Sym}^{m_{1}}f_{1}$ and
$\pi^{\prime}=\operatorname{Sym}^{m_{2}}f_{2}$. Theorem 9.1 implies that for
$0<\theta<\frac{2}{2(m_{1}+1)(m_{2}+1)+1}$, one has for any $A>0$ that

$\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
(q,q_{\pi}q_{\pi^{\prime}})=1\end{subarray}}\max_{\begin{subarray}{c}y\leq
x\\\
(a,q)=1\end{subarray}}|\psi_{\operatorname{Sym}^{m_{1}}f_{1}\times\operatorname{Sym}^{m_{2}}f_{2}}(y;q,a)|\ll\frac{x}{(\log
x)^{A}}$

where the implied constant only depends on $f_{1},f_{2},m_{1},m_{2},\theta,A$.
Using (7.2) and (8.3), a standard exercise in partial summation yields the
result. ∎

###### Proof of Theorem 3.3.

We have

$\displaystyle\sum_{\begin{subarray}{c}x<p\leq 2x\\\ p\equiv a\ (\mathrm{mod}\
q)\end{subarray}}F(\theta_{1}(p),\theta_{2}(p))-\widehat{F}(0,0)\frac{\pi(2x)-\pi(x)}{\varphi(q)}$
$\displaystyle=\widehat{F}(0,0)\Big{(}\sum_{\begin{subarray}{c}x<p\leq 2x\\\
p\equiv a\ (\mathrm{mod}\
q)\end{subarray}}1-\frac{\pi(2x)-\pi(x)}{\varphi(q)}$ $\displaystyle\Big{)}$
$\displaystyle+\sum_{\begin{subarray}{c}0\leq m_{1}\leq M_{1}\\\ 0\leq
m_{2}\leq M_{2}\\\
(m_{1},m_{2})\neq(0,0)\end{subarray}}\widehat{F}(m_{1},m_{2})$
$\displaystyle\sum_{\begin{subarray}{c}x<p\leq 2x\\\ p\equiv a\ (\mathrm{mod}\
q)\end{subarray}}U_{m_{1}}(\cos\theta_{1}(p))U_{m_{2}}(\cos\theta_{2}(p)).$

Taking the maximum over $(a,q)=1$, summing over $q\leq x^{\theta}$ with
$(q,N_{1}N_{2})=1$ and then applying the triangle inequality, we can use the
classical Bombieri–Vinogradov theorem to bound the first term and Theorem 9.5
to bound the second term, completing the proof. ∎

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