# Chebotarev Density Theorem and extremal primes for non-CM elliptic curves

Amita Malik  American Institute of Mathematics, San Jose, California 95112
<EMAIL_ADDRESS>and  Neha Prabhu  Indian Institute of Science Education
and Research Pune, Dr. Homi Bhabha Road, Pune - 411008, Maharashtra, India
<EMAIL_ADDRESS>
###### Abstract.

For a fixed non-CM elliptic curve $E$ over $\mathbb{Q}$ and a prime $\ell$, we
prove an asymptotic formula on the number of primes $p\leq x$ for which the
Frobenius trace $a_{p}(E)$ satisfies the congruence
$a_{p}(E)\equiv[2\sqrt{p}]\pmod{\ell}$. In order to achieve this, we establish
a joint distribution concerning the fractional part of $\alpha p^{\theta}$ for
$\theta\in[0,1],\ \alpha>0$, and primes $p$ satisfying the Chebotarev
condition. As a corollary, we also obtain upper bounds for the number of
extremal primes. The results rely on GRH for Dedekind zeta functions for
Galois extensions of number fields.

###### Key words and phrases:

Non-CM Elliptic curves, Extremal primes, Lang-Trotter conjecture, Chebotarev
Density Theorem, Joint distribution, Fractional part of powers of primes

###### 2010 Mathematics Subject Classification:

11G05, 11N05

##  1\. Introduction

Let $E$ denote an elliptic curve over $\mathbb{Q}$ without complex
multiplication with conductor $N_{E}$. For a prime $p$ of good reduction, $E$
reduces to an elliptic curve over the finite field $\mathbb{F}_{p}$ and we
denote by $a_{p}(E)$ the trace of the Frobenius automorphism acting on the
points of $E$ over ${\overline{\mathbb{F}}}_{p}$. Then,
$a_{p}(E)=p+1-\\#E(\mathbb{F}_{p})$, and the Hasse bound $|a_{p}(E)|\leq
2\sqrt{p}$ holds. The last few decades have seen much research pertaining to
the distribution of the sequence $a_{p}(E)/2\sqrt{p}$. Sato and Tate
formulated a conjecture for the distribution of this sequence in the interval
$[-1,1]$ around 1960 which was proven by Taylor, Clozel, Harris and Shepherd-
Barron in [Tay08, CHT08, HSBT10].

######  Theorem (Sato-Tate conjecture).

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$ with conductor ${N}_{E}$.
Let $\alpha,\beta\in\mathbb{R}$ with $-1\leq\alpha\leq\beta\leq 1$. Then, as
$x\rightarrow\infty$,

$\frac{1}{\pi(x)}\\#\left\\{p\leq
x,p\nmid{N}_{E}\;:\;\frac{a_{p}(E)}{2\sqrt{p}}\in(\alpha,\beta)\right\\}\sim\frac{2}{\pi}\int_{\alpha}^{\beta}\sqrt{1-t^{2}}~{}dt.$

The measure
$\mu_{ST}([\alpha,\beta]):=\frac{2}{\pi}\int_{\alpha}^{\beta}\sqrt{1-t^{2}}~{}dt$
is also known as the semicircle measure. The shape of the distribution clearly
suggests fewer primes at the ends of the interval in comparison to those at
the middle, so it is interesting to see if a precise statement can be made
about the behaviour at the extremes. The objects of our study here are such
primes, called extremal primes i.e. primes $p$ satisfying
$a_{p}(E)=\pm[2\sqrt{p}]$, where $[\cdot]$ denotes the greatest integer
function. These were first studied by James et al. [JTT+16] (see also [JP17])
who conjectured that, as $x\to\infty$,

$\displaystyle\\#\left\\{p\leq x:p\nmid
N_{E},a_{p}(E)=\pm[2\sqrt{p}]\right\\}$ (1.1) $\displaystyle\sim$
$\displaystyle\begin{cases}\displaystyle\frac{2}{3\pi}\;\frac{x^{3/4}}{\log{x}}&\mbox{if
$E$ has complex multiplication (CM);}\vspace{5pt}\\\
\displaystyle\frac{8}{3\pi}\;\frac{x^{1/4}}{\log{x}}&\mbox{if $E$ does not
have CM.}\end{cases}$

The number of extremal primes is smaller for the case of non-CM curves since
the measure of the interval at the ends of the Sato-Tate semi-circle
distribution is smaller. On the other hand, for CM curves there is an excess
of such primes since the measure for the distribution of
$a_{p}(E)/(2\sqrt{p})$ in $[\alpha,\beta]\subseteq[-1,1]/\\{0\\}$ is given by
$\frac{1}{2\pi}\int_{\alpha}^{\beta}\frac{dt}{\sqrt{1-t^{2}}}.$ This is
reminiscent of the Lang-Trotter conjecture where the trace is a fixed integer
rather than being a function of the prime. More precisely,

######  Conjecture 1.1 (Lang-Trotter conjecture).

$\\#\left\\{p\leq x:p\nmid N_{E},\,a_{p}(E)=t\right\\}\sim
C_{E,t}\frac{x^{1/2}}{\log{x}}$

as $x\to\infty$, where $t\in\mathbb{Z}$ is fixed and $C_{E,t}$ is a constant
depending on the curve $E$ and the integer $t$. This conjecture is far from
being proved and thus upper bounds remain of great interest.

The asymptotics in (1.1) for extremal primes for CM curves were proven by
James and Pollack [JP17]. In a subsequent paper, Agwu, Harris, James, Kannan
and Li [AHJ+18] studied asymptotic behavior of the primes where $a_{p}(E)$
falls within a small range of the end of the Hasse interval. In the case of
non-CM curves, the asymptotic was proved to be true on average in the Ph. D.
thesis of Giberson [Gib17, GJ18]. In contrast, the asymptotics (1.1) for a
fixed non-CM curve seem to be out of reach with current techniques. In
[DGM+20], along with David, Gafni, and Turnage-Butterbaugh, the authors
established the following upper bounds for a single curve $E/\mathbb{Q}$.

######  Theorem ([DGM+20]).

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assume that for $n\geq 0$, the
$L$-functions $L(s;Sym^{n}(E))$ have analytic continuation to the entire
complex plane (except for a simple pole at $s=1$ when $n=0$), satisfy the
expected functional equation and the Generalized Riemann Hypothesis (GRH).
Then

$\\#\\{x<p\leq 2x:p\nmid N_{E},\;a_{p}(E)=[2\sqrt{p}]\\}\ll_{E}{x^{1/2}}.$
(1.2)

######  Remark 1.

Due to the recent breakthrough of Newton and Thorne [NT20], the only unproven
hypothesis in the above result is GRH for symmetric powers of $L$-functions.

More recently, from Gafni, Thorner and Wong [GTW20] it follows that the bound
in (1.2) can be made $x(\log\log x)^{2}/(\log x)^{2}$ unconditionally.

In this article, we follow a different approach to count extremal primes.
Inspired by the work of Serre [Ser72] and Murty, Murty and Saradha [MMS88] on
bounds for the Lang-Trotter conjecture, we investigate the number of primes up
to $x$ satisfying the extremality condition modulo a large prime $\ell$. The
distribution of $a_{p}(E)$ modulo $\ell$ is known by the Chebotarev Density
Theorem applied to the Galois extensions $\mathbb{Q}(E[\ell])/\mathbb{Q}$, the
field obtained by adjoining the coordinates of $\ell$-torsion points of $E$ to
$\mathbb{Q}$. A novelty in our application is the observation that studying
primes $p$ at the end of the Sato-Tate distribution leads to studying the
distribution of the fractional part of $2\sqrt{p}$. Balog [Bal83] showed that
the latter distribution is uniform. For our purpose, we require the joint
distribution of rational primes satisfying a Chebotarev condition in the
extensions $\mathbb{Q}(E[\ell])/\mathbb{Q}$ with this fractional part lying in
certain interval. Most of the paper is devoted to proving this result for
arbitrary finite Galois extensions over $\mathbb{Q}$, so as to be of
independent interest.

We also give a version of this joint distribution in the case when the
fractional part $\\{\sqrt{p}\\}<p^{-\lambda}$ for $\lambda<1/4$. Note that for
$\lambda=1/2$, this concerns the infinitude of primes $p$ of the form
$n^{2}+1$ for some $n\in\mathbb{N}$, also known as one of Landau’s four
problems and is widely open.

Before stating our main results, we fix some notations. Throughout the paper,
$p$ and $\ell$ denote primes, $\pi(x)$ denotes the number of primes up to $x$,
and $\\{y\\},[y]$ denote the fractional part and the integer part of $y$,
respectively. For a finite Galois extension $L/\mathbb{Q}$ and $C$ a union of
conjugacy classes in $\mbox{Gal}(L/\mathbb{Q})$, define

$\displaystyle\pi_{C}(x,L)$
$\displaystyle:=\pi_{C}(x,L/\mathbb{Q}):=\\#\\{x<p\leq 2x:p\text{ unramified
in }L,\sigma_{p}\in C\\}$

where $\sigma_{p}$ is the conjugacy class of the Frobenius automorphism
associated with any prime lying above $p$. To simplify exposition, henceforth
when we write $\sigma_{p}\in C$ we assume that $p$ is unramified in $L$ unless
mentioned otherwise. We now state our main results.

######  Theorem 1.1.

Consider a finite Galois extension $L/\mathbb{Q}$ with $n_{L}=[L:\mathbb{Q}]$.
Let $\alpha>0$, $\omega\geq 1$ and $0\leq\delta_{1}<\delta_{2}\leq 1$ with
$\delta:=\delta_{2}-\delta_{1}$. Let $\theta\in[0,1]$ be fixed. Assume that
GRH holds for the Dedekind zeta function $\zeta_{L}(s)$ and the parameters
$n_{L},\alpha,\omega$ and $\delta$ satisfy

$\alpha^{1/4}(\omega n_{L}/\delta)^{1/2}(\log
x)^{2}\ll_{\theta}x^{(1-\theta)/4}.$

Then the following holds uniformly for $\delta,\alpha$ and $\omega$.

$\displaystyle\\#\\{x<p\leq 2x:\,\delta_{1}\leq\\{\alpha
p^{\theta}\\}<\delta_{2}\text{ and }\sigma_{p}\in C\\}-\delta\pi_{C}(x,L)$
$\displaystyle\quad\ll_{\theta}\frac{|C|}{|G|}n_{L}\log
x\left(\frac{(\delta\omega)^{1/2}\alpha^{1/4}}{n_{L}^{1/2}}x^{(3+\theta)/4}+\frac{\delta\omega}{\alpha^{1/2}}x^{1-\theta/2}\log
x\right)$ $\displaystyle\quad+(\delta
n_{L}\omega)^{1/2}\alpha^{1/4}x^{(1+\theta)/4}\log
x+\frac{|C|}{|G|}\frac{\delta x}{\omega\log x}.$

######  Theorem 1.2.

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assume that GRH holds for
$\zeta_{\mathbb{Q}(E[\ell])}(s)$, where $\ell$ is a large prime. Then, for
$\ell\ll x^{1/18}\omega^{-2/9}\log^{-8/9}x$, $\omega\geq 1$,

$\displaystyle\\#\left\\{x<p\leq 2x:p\nmid
N_{E},\;a_{p}(E)\equiv[2\sqrt{p}]\bmod\ell\right\\}=\frac{\pi(x)}{\ell}$
$\displaystyle\quad+\operatorname{O}_{E}\left(\frac{x}{\omega\ell\log
x}+\omega^{1/2}\ell^{5/4}x^{7/8}\log x+\omega\ell^{7/2}x^{3/4}(\log
x)^{2}\right).$

Making the simple observation that

$\displaystyle\\#\left\\{x<p\leq 2x:p\nmid
N_{E},a_{p}(E)=[2\sqrt{p}]\right\\}$ $\displaystyle\quad\leq\\#\left\\{x<p\leq
2x:p\nmid N_{E},a_{p}(E)\equiv[2\sqrt{p}]\bmod\ell\right\\}$

and setting $\omega=1$ and $\displaystyle{\ell=x^{1/18}\log^{-8/9}x}$ in
Theorem 1.2, one obtains the following upper bounds for the number of extremal
primes up to $x$.

######  Corollary 1.

Assume GRH as in Theorem 1.2. For a non-CM elliptic curve $E/\mathbb{Q}$, and
large $x$,

$\left\\{x<p\leq 2x:p\nmid
N_{E},\;a_{p}(E)=[2\sqrt{p}]\right\\}\ll_{E}x^{17/18}(\log x)^{-1/9}.$

######  Remark 2.

Similar results can be obtained for extremal primes with
$a_{p}(E)=-[2\sqrt{p}]$ using essentially the same arguments as presented
here. Moreover, because of the generality in Theorem 1.1, one can write more
general versions of Theorem 1.2 and Corollary 1 where $\sqrt{p}$ is replaced
by $p^{\theta}$ for $\theta\in[0,1]$.

As in the case of upper bounds for the Lang-Trotter conjecture, we obtain
better estimates for the joint distribution in the particular case of
$a_{p}(E)\equiv 0\mod\ell$. To be precise, the following holds.

######  Corollary 2.

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$. Assume that GRH holds
for $\zeta_{\mathbb{Q}(E[\ell])}(s)$ where $\ell$ is a large prime with
$\ell\ll x^{1/14}(\log x)^{-8/7}\omega^{-2/7}$. Then

$\displaystyle\\#\left\\{x<p\leq 2x:p\nmid
N_{E},\;a_{p}(E)\equiv[2\sqrt{p}]\equiv
0\bmod\ell\right\\}=\frac{\pi(x)}{\ell^{2}}$
$\displaystyle\quad+\operatorname{O}\left(\frac{x}{\ell^{2}\omega\log
x}+\frac{\omega^{1/2}x^{7/8}\log x}{\ell^{1/4}}+\omega\ell^{3/2}x^{3/4}(\log
x)^{2}\right).$

In fact, using the same ideas as in Theorem 1.1, one can prove a more general
result, stated below, where the bounds for the fractional part of the prime
are themselves a function of the prime.

######  Theorem 1.3.

Let $\alpha,\lambda>0$ and $\theta\in[0,1]$ be fixed. For finite Galois
extension $L/K$, assume GRH for $\zeta_{L}(s)$. Then

$\displaystyle\\#\\{x<p\leq 2x:\\{\alpha p^{\theta}\\}<p^{-\lambda}\text{ and
}\sigma_{p}\in C\\}-\sum_{\begin{subarray}{c}x<p\leq 2x\\\ \sigma_{p}\in
C\end{subarray}}p^{-\lambda}$
$\displaystyle\ll\frac{|C|}{|G|}\frac{x^{1-\lambda}}{\omega\log
x}+\omega{\alpha^{1/2}}\ x^{\theta/2}\log d_{L}\log^{3}x$
$\displaystyle\quad+\frac{|C|}{|G|}{\alpha^{1/2}}\
x^{(1+\theta)/2}\log^{3}x\left(\log d_{L}+n_{L}\log
x\right)\left(\omega^{2}+\frac{\omega
x^{1/2-\theta-\lambda}}{\alpha^{1/2}}\right),$

where $\omega\geq 1$ is a parameter at our disposal.

Setting $\theta=1/2,\alpha=1$ and $\lambda=1/4-\epsilon$ for any $\epsilon>0$,
we obtain the following asymptotic result, in spirit of the Landau’s prime
counting problem where the primes satisfy $\\{\sqrt{p}\\}<p^{-1/2}$ i.e. $p-1$
is a perfect square.

######  Corollary 3.

For a finite Galois extension $L/\mathbb{Q}$, assume that GRH holds for
$\zeta_{L}(s)$. Then, for a fixed $\epsilon>0$

$\displaystyle\\#\\{x<p\leq 2x:\\{\sqrt{p}\\}<p^{-1/4+\epsilon}\text{ and
}\sigma_{p}\in C\\}-\sum_{\begin{subarray}{c}x<p\leq 2x\\\ \sigma_{p}\in
C\end{subarray}}p^{-1/4+\epsilon}$
$\displaystyle\ll\frac{|C|}{|G|}\frac{x^{3/4+\epsilon}}{\omega\log x}+\omega
x^{1/4}\log d_{L}\log^{3}x+\frac{|C|}{|G|}\omega^{2}x^{3/4}\log^{3}x\left(\log
d_{L}+n_{L}\log x\right).$

Setting $\alpha=1,\theta=1/2$, and $\delta_{1}=0$ in Theorem 1.1, one obtains
the following (conditional) generalization of [Bal85] that investigates the
distribution of fractional parts of $p^{\theta}$ with $p\equiv a\bmod q$.

######  Corollary 4.

Assume the notations and hypotheses as in Theorem 1.1. Then

$\displaystyle\\#\\{x<p\leq 2x:\\{\sqrt{p}\\}<\delta\text{ and }\sigma_{p}\in
C\\}-\delta\pi_{C}(x,L)$ $\displaystyle\ll\frac{|C|}{|G|}\left((\delta\omega
n_{L})^{1/2}x^{7/8}\log x+n_{L}\delta\omega x^{3/4}(\log x)^{2}+\frac{\delta
x}{\omega\log x}\right).$

Theorem 1.1 is also of a similar flavour as some other interesting joint
distribution theorems such as asymptotics for the number of Charmichael
numbers composed of primes satisfying the Chebotarev condition studied in
[BGY13]. In [AG15], an asymptotic estimate for the number of Piatetski-Shapiro
primes up to $x$ satisfying the Chebotarev condition is derived.

######  Remark 3.

One can also write versions of Theorems 1.1 and 1.3 for general extensions
$L/K$ where $K$ is a finite extension of $\mathbb{Q}$ and $L$ is a normal
extension of $K$ with Galois group $G=G(L/K)$. For a fixed conjugacy class $C$
of $G$, here one would count primes $\mathfrak{p}\in K$ of norm
$N_{K/\mathbb{Q}}\mathfrak{p}\leq x$ which are unramified in $L$ such that the
Artin symbol satisfies $[\frac{L/K}{\mathfrak{p}}]=C$ and simultaneously have
the fractional part $\\{\alpha(N_{K/\mathbb{Q}}\mathfrak{p})^{\theta}\\}$
lying in the interval $[\delta_{1},\delta_{2})$ of length $\delta$. The main
term would then be $\delta\pi_{C}(x,L/K)$ and the error terms can be shown to
be as in Theorems 1.1 and 1.3.

The structure of this paper is as follows. We begin Section 2 with some
preliminaries and then prove Theorem 1.2 assuming Theorem 1.1. In Section 3,
we first prove Theorem 1.1 adapting the ideas of Balog [Bal83] and Lagarias-
Odlyzko [LO77]. The proofs of Theorems 1.3 and Corollary 2 are also presented
in the same section. Lastly, in Section 4, we provide details of the results
needed in the proof of Theorem 1.1.

Acknowledgements: The authors are grateful to Chantal David for suggesting the
problem and for many fruitful discussions related to the paper. The first
author is supported by NSF Grant DMS-1854398.

##  2\. Preliminaries and proof of Theorem 1.2

Let $N_{E}$ denote the conductor of the elliptic curve (without complex
multiplication) over $\mathbb{Q}$. For a given prime $\ell$, let $E[\ell]$
denote the $\ell$-torsion points subgroup of $E[\bar{\mathbb{Q}}]$. It is
known that the Galois representation

$\rho_{E,\ell}:\mbox{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mbox{Aut}_{\mathbb{F}_{\ell}}(E[\ell])\cong\mbox{GL}_{2}(\mathbb{F}_{\ell})$

is unramified at all primes $p\nmid N_{E}\ell$. The field
$\mathbb{Q}(E[\ell])$, obtained by adjoining the coordinates of all the
$\ell$-torsion points of $E$ to $\mathbb{Q}$, is the fixed field in
$\bar{\mathbb{Q}}$ of ker$\rho_{E,\ell}$. Serre [Ser72] showed that for all
but finitely many primes $\ell$, the representation $\rho_{E,\ell}$ is
surjective. Thus, using $\rho_{E,\ell}$ we see that for all but finitely many
primes $\ell$, the Galois group
$G_{\ell}:=\mbox{Gal}(\mathbb{Q}(E[\ell])/\mathbb{Q})$ is equal to
$\mbox{GL}_{2}(\mathbb{F}_{\ell})$. Moreover, the characteristic polynomial of
$\rho_{E,\ell}(\sigma_{p})$ is given by

$x^{2}-a_{p}(E)x+p\,(\bmod\ell).$

That is, $a_{p}(E)$ is the trace of the Frobenius automorphism at $p$.
Therefore, for $a\in\mathbb{F}_{\ell}$, if $C_{\ell}(a)$ denotes the union of
conjugacy classes in $\mbox{GL}_{2}(\mathbb{F}_{\ell})$ of elements of trace
$a$ modulo $\ell$, then

$a_{p}(E)\equiv a\bmod\ell\iff\sigma_{p}\in C_{\ell}(a).$

We now review the structure of conjugacy classes in
$\mbox{GL}_{2}(\mathbb{F}_{\ell})$ for an odd prime $\ell$. This is well
known, see for example [FH91, Section 5.2] and is listed below for the
convenience of the reader.

Class representative |  No. of classes |  Size of class
---|---|---
$\begin{pmatrix}\lambda&0\\\ 0&\lambda\end{pmatrix}$, $\lambda\in\mathbb{F}_{\ell}^{\times}$ |  $\ell-1$ |  $1$
$\begin{pmatrix}\lambda&1\\\ 0&\lambda\end{pmatrix}$, $\lambda\in\mathbb{F}_{\ell}^{\times}$ |  $\ell-1$ |  $\ell^{2}-1$
$\begin{pmatrix}\lambda_{1}&0\\\ 0&\lambda_{2}\end{pmatrix}$, $\lambda_{1}\neq\lambda_{2}$, $\lambda_{1},\lambda_{2}\in\mathbb{F}_{\ell}^{\times}$ |  $\frac{1}{2}(\ell-1)(\ell-2)$ |  $\ell(\ell+1)$
$\begin{pmatrix}\alpha&D\beta\\\ \beta&\alpha\end{pmatrix}$, $\alpha,\beta\in\mathbb{F}_{\ell},\beta\neq 0$, $D$ is a fixed non-square in $\mathbb{F}_{\ell}^{\times}$ |  $\frac{\ell}{2}(\ell-1)$ |  $\ell(\ell-1)$

Therefore,

$\displaystyle\frac{|C_{\ell}(a)|}{|G_{\ell}|}$ $\displaystyle=$
$\displaystyle\begin{cases}\displaystyle\frac{\ell^{2}-\ell-1}{(\ell-1)^{2}(\ell+1)}&\mbox{for
$a\neq 0$}\vspace{5pt}\\\ \displaystyle\frac{\ell}{(\ell-1)(\ell+1)}&\mbox{for
$a=0$}\end{cases}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell}+\operatorname{O}\left(\frac{1}{\ell^{2}}\right).$

Moreover, for $a\neq 0$, $C_{\ell}(a)$ is a union of $\ell$ conjugacy classes,
while $C_{\ell}(0)$ is a union of $\ell-1$ conjugacy classes. We use the
following effective Chebotarev density theorem of Lagarias-Odlyzko [LO77]
which is sufficient for our purposes.

######  Theorem ([LO77]).

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$
and $C$ be a conjugacy class in $G$. There exists an effectively computable
positive absolute constant $c_{1}$ such that if $\zeta_{L}(s)$ satisfies the
GRH, then for every $x\geq 2$

$\pi_{C}(x,L/K)-\frac{|C|}{|G|}\pi(x)\leq c_{1}\frac{|C|}{|G|}x^{1/2}\log
d_{L}x^{n_{L}}+\log d_{L}.$ (2.2)

We now prove Theorem 1.2.

###### Proof.

For each residue $a\in\left\\{0,1,\ldots,\ell-1\right\\}$, we have

$[2\sqrt{p}]\equiv
a\bmod\ell\iff\left\\{\frac{2\sqrt{p}}{\ell}\right\\}\in\left[\frac{a}{\ell},\frac{a+1}{\ell}\right)$

and

$a_{p}(E)\equiv a\bmod\ell\iff\sigma_{p}\in C_{\ell}(a)\subseteq G_{\ell}.$

Therefore, by Theorem 1.1 with $\theta=1/2,\alpha=2/\ell,$ and
$\delta=1/\ell$, we conclude

$\displaystyle\\#$ $\displaystyle\left\\{p\leq
x\;:\;a_{p}(E)\equiv[2\sqrt{p}]\equiv a\bmod\ell\right\\}=\frac{x}{\ell\log
x}$
$\displaystyle\quad+\operatorname{O}\left(\omega^{1/2}\ell^{1/4}x^{7/8}\log
x+\frac{x}{\omega\ell^{2}\log x}\right).$

This gives us,

$\displaystyle\\#$ $\displaystyle\left\\{p\leq
x\;:\;a_{p}(E)\equiv[2\sqrt{p}]\bmod\ell\right\\}$
$\displaystyle=\sum_{a\bmod\ell}\\#\left\\{p\leq x\;:\;\sigma_{p}\in
C_{\ell}(a)\;\text{and}\;\left\\{\frac{2\sqrt{p}}{\ell}\right\\}\in\left[\frac{a}{\ell},\frac{a+1}{\ell}\right)\right\\}$
$\displaystyle=\sum_{a\bmod\ell}\frac{1}{\ell}\
\pi_{C_{\ell}(a)}\left(x,\mathbb{Q}(E[\ell])\right)+\operatorname{O}\left(x^{7/8}\ell^{1/4}\log
x+\frac{x}{\omega\ell^{2}\log x}\right)$
$\displaystyle=\pi_{C_{\ell}(a)}\left(x,\mathbb{Q}(E[\ell])\right)+\operatorname{O}\left(\omega^{1/2}\ell^{5/4}x^{7/8}\log
x+\frac{x}{\omega\ell\log x}\right),$

where we have used (2) and (2.2) to compute
$\pi_{C_{\ell}(a)}\left(x,\mathbb{Q}(E[\ell])\right)$. ∎

##  3\. The joint distribution theorem and some applications

Remark. In what follows, we may assume that the parameters $\alpha$ and
$\theta$ satisfy $\alpha x^{\theta}\geq 1$ since for $\alpha x^{\theta}<1$,
$\alpha x^{\theta}=\\{\alpha x^{\theta}\\}\in[\delta_{1},\delta_{2})$, and
therefore the desired quantity can be computed using the Prime Number Theorem.

###  3.1. Proof of Theorem 1.1

###### Proof.

To start with, we write the fractional part condition as follows:

$\displaystyle[\alpha p^{\theta}-\delta_{1}]-[\alpha p^{\theta}-\delta_{2}]$
$\displaystyle=$ $\displaystyle\begin{cases}1&\mbox{if
$\delta_{1}\leq\\{\alpha p^{\theta}\\}<\delta_{2}$;}\\\
0&\mbox{otherwise.}\end{cases}$

First, we obtain the result when $x_{j}<p\leq x_{j+1}$ for
$x_{j}:=[x(1+j/B)]+1/2$ for $j=0,\ldots,B$ and $B=[\omega]$. Summing over $j$
then establishes the result for $x<p\leq 2x$.

With $\delta=\delta_{2}-\delta_{1}$, the length of the interval, we set

$\displaystyle U_{-}$ $\displaystyle:=$ $\displaystyle\frac{\alpha
x_{j}^{\theta}}{\delta},\qquad U_{+}:=\frac{\alpha x_{j+1}^{\theta}}{\delta}.$

Then for $x_{j}<p\leq x_{j+1}$, we have

$\alpha p^{\theta}\left(1-\frac{1}{U_{-}}\right)-\delta_{1}\leq\alpha
p^{\theta}-\delta_{2}\leq\alpha
p^{\theta}\left(1-\frac{1}{U_{+}}\right)-\delta_{1}.$

Note that we are interested in the sum

$S:=\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}[\alpha p^{\theta}-\delta_{1}]-[\alpha p^{\theta}-\delta_{2}]$

in order to bound the number of primes $x_{j}<p\leq x_{j+1}$ such that
$\sigma_{p}\in C$ and $\delta_{1}\leq\\{\alpha p^{\theta}\\}<\delta_{2}.$ This
implies

$\displaystyle S$ $\displaystyle\geq$
$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\left[\alpha p^{\theta}-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{+}}\right)-\delta_{1}\right]$

and

$\displaystyle S$ $\displaystyle\leq$
$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\left[\alpha p^{\theta}-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{-}}\right)-\delta_{1}\right].$

Therefore, using

$\frac{\alpha
p^{\theta}}{U_{\pm}}=\delta+\operatorname{O}\left(\frac{\delta}{\omega}\right),$

in order to obtain the claimed asymptotics for $S$, it suffices to prove

$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\left[\alpha p^{\theta}-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]-\frac{\alpha
p^{\theta}}{U_{\pm}}$ $\displaystyle\ll\frac{|C|}{|G|}\log x\left((\delta
n_{L}/\omega)^{1/2}\alpha^{1/4}x^{(3+\theta)/4}+\frac{\delta
n_{L}}{\alpha^{1/2}}x^{1-\theta/2}\log x\right)$ $\displaystyle\quad+(\delta
n_{L}/\omega)^{1/2}\alpha^{1/4}x^{(1+\theta)/4}\log x.$ (3.1)

We may write the summand

$\displaystyle\left[\alpha p^{\theta}-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]$
$\displaystyle=\sum_{m\leq\alpha p^{\theta}-\delta_{1}}1\ -\sum_{m\leq\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}}1.$ (3.2)

Define the sequence $(a_{m})_{m\in\mathbb{N}}$

$a_{m}:=\begin{cases}1&\text{ if }\frac{1}{3}\alpha
x^{\theta}-\delta_{1}<m\leq 3\alpha x^{\theta}-\delta_{1}\\\ 0&\text{
otherwise}.\end{cases}$

Set

$A_{\delta_{1}}(M):=\sum_{m\leq M-\delta_{1}}a_{m}=\sum_{m\geq 1}a_{m}\
f\left(\frac{m+\delta_{1}}{M}\right)$

where

$f(y)=\begin{cases}1&\text{ if }0<y<1\\\ 1/2&\text{ if }y=1\\\ 0&\text{ if
}y>1\end{cases}$

Using the inverse Mellin transform, for $\sigma>0$, we write

$f(y)=\frac{1}{2\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{y^{-s}}{s}~{}ds.$

For $y=(m+\delta_{1})/M$, this gives

$A_{\delta_{1}}(M)=\frac{1}{2\pi
i}\int_{\sigma-i\infty}^{\sigma+i\infty}\sum_{m\geq
1}\frac{a_{m}}{(m+\delta_{1})^{s}}\frac{M^{s}}{s}~{}dx.$ (3.3)

Thus, using the definition of the sequence $(a_{m})$, we rewrite (3.2) as

$\displaystyle\left[\alpha p^{\theta}-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]$
$\displaystyle=\sum_{m\leq\alpha p^{\theta}-\delta_{1}}a_{m}\
-\sum_{m\leq(\alpha
p^{\theta})\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}}a_{m}$
$\displaystyle=A_{\delta_{1}}(\alpha p^{\theta})-A_{\delta_{1}}\left(\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)\right).$

In order to estimate the integrals $A_{\delta_{1}}(\alpha p^{\theta})$ and
$A_{\delta_{1}}\left(\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)\right)$, given by (3.3), we make
use of the truncated Perron’s formula [Tit86, Lemma 3.12], and obtain

$\displaystyle\left[\alpha p^{\theta}-\delta_{1}\right]-\left[(\alpha
p^{\theta})\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]=\frac{1}{2\pi
i}\int_{1/2-iT_{1}}^{1/2+iT_{1}}L(s)H(s)\ p^{\theta s}~{}ds$
$\displaystyle\quad+\operatorname{O}\Bigg{(}\sum_{\frac{1}{3}\alpha
x^{\theta}<m+\delta_{1}\leq 3\alpha
x^{\theta}}\\!\min\left\\{1,T_{1}^{-1}\left|\log\frac{\alpha
p^{\theta}}{m+\delta_{1}}\right|^{-1}\right\\}\Bigg{)}$
$\displaystyle\quad+\operatorname{O}\Bigg{(}\sum_{\frac{1}{3}\alpha
x^{\theta}<m+\delta_{1}\leq 3\alpha
x^{\theta}}\\!\min\left\\{1,T_{1}^{-1}\left|\log\frac{\alpha
p^{\theta}}{m+\delta_{1}}\left(1-\frac{1}{U_{\pm}}\right)\right|^{-1}\right\\}\Bigg{)}$
$\displaystyle=\frac{1}{2\pi i}\int_{1/2-iT_{1}}^{1/2+iT_{1}}L(s)H(s)p^{\theta
s}~{}ds+\operatorname{O}\Bigg{(}\frac{\alpha x^{\theta}}{T_{1}}\log(\alpha
x^{\theta})\Bigg{)},$

where

$H(s):=\frac{1}{s}\left(1-\left(1-\frac{1}{U_{\pm}}\right)^{s}\right)\ll\frac{1}{U_{\pm}}$

and

$L(s):=\alpha^{s}\sum_{\frac{1}{3}\alpha x^{\theta}-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}\frac{1}{(m+\delta_{1})^{s}}.$

Summing over the primes, we find

$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\Bigg{(}\left[\alpha p^{\theta}\
-\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]\Bigg{)}$
$\displaystyle\hskip 18.06749pt=\frac{1}{2\pi
i}\int_{1/2-iT_{1}}^{1/2+iT_{1}}L(s)H(s)F(-\theta
s)~{}ds+\operatorname{O}\left(\frac{\alpha x^{\theta}}{T_{1}}\log(\alpha
x^{\theta})\pi_{C}(x_{j},L)\right)$ (3.4)

where

$\displaystyle F(s):=\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\
\sigma_{p}\in C\end{subarray}}p^{-s}\qquad\text{ and
}\qquad\pi_{C}(x_{j},L):=\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\
\sigma_{p}\in C\end{subarray}}1.$

First, we compute the above integral in the smaller range, up to
$T_{0}:=\alpha x^{\theta}$. Observe that in the range $|t|\leq T_{0}$, we have

$\displaystyle
H(s)=\frac{1}{U_{\pm}}+\operatorname{O}\left(\frac{|s-1|}{U_{\pm}^{2}}\right)$

and

$L(s)=\alpha^{s}\frac{(3\alpha x^{\theta})^{1-s}-(\alpha
x^{\theta}/3)^{1-s}}{1-s}+\operatorname{O}\left(x^{-\theta\Re(s)}\right)$

Therefore, for the integral in (3.4) in the range $|t|\leq T_{0}$, we have

$\displaystyle\frac{1}{2\pi i}\int_{1/2-iT_{0}}^{1/2+iT_{0}}L(s)H(s)F(-\theta
s)\,ds$ $\displaystyle=\frac{1}{U_{\pm}}\sum_{\begin{subarray}{c}x_{j}<p\leq
x_{j+1}\\\ \sigma_{p}\in C\end{subarray}}\alpha p^{\theta}\;\frac{1}{2\pi
i}\int_{1/2-iT_{0}}^{1/2+iT_{0}}(\alpha p^{\theta})^{s-1}\frac{(3\alpha
x^{\theta})^{1-s}-(\alpha x^{\theta}/3)^{1-s}}{1-s}~{}ds$ (3.5)
$\displaystyle\quad+\operatorname{O}\left(\frac{\delta}{\alpha}x^{-3\theta/2}\int_{-T_{0}}^{T_{0}}|F(-\theta/2-i\theta
t)|\;dt\right).$ (3.6)

Using Cauchy-Schwarz inequality, the error term (3.6) is bounded by

$\displaystyle\ll\frac{\delta}{\alpha}x^{-3\theta/2}T_{0}^{1/2}\left(\int_{-T_{0}}^{T_{0}}|F(-\theta/2-i\theta
t)|^{2}~{}dt\right)^{1/2}.$ (3.7)

Also, by applying the mean value theorem for Dirichlet polynomials, we have

$\int_{-T_{0}}^{T_{0}}\left|F(-\theta/2-i\theta t)\right|^{2}~{}dt\ll
x^{\theta}\pi_{C}(x_{j},L)\;\left(T_{0}+x/\omega\right).$

Inserting this estimate in (3.7), the error term (3.6) is bounded by

$\displaystyle\ll\delta T_{0}^{1/2}(\alpha
x^{\theta})^{-1}\pi_{C}(x_{j},L)^{1/2}\left(T_{0}^{1/2}+x^{1/2}/\omega^{1/2}\right)$
$\displaystyle\ll\delta\pi_{C}(x_{j},L)^{1/2}\left(1+x^{(1-\theta)/2}(\alpha\omega)^{-1/2}\right),$
(3.8)

using $T_{0}=\alpha x^{\theta}$. We now compute the integral in (3.5) to
obtain the desired main term. The change of variable $w=1-s$ yields

$\displaystyle\frac{1}{U_{\pm}}\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\
\sigma_{p}\in C\end{subarray}}\alpha p^{\theta}\frac{1}{2\pi
i}\int_{1/2-iT_{0}}^{1/2+iT_{0}}\frac{1}{w}\left(\left(\frac{3x^{\theta}}{p^{\theta}}\right)^{w}-\left(\frac{x^{\theta}}{3p^{\theta}}\right)^{w}\right)~{}dw.$

By Perron’s formula, for all values $x_{j}<p\leq x_{j+1}$,

$\frac{1}{2\pi
i}\int_{1/2-iT_{0}}^{1/2+iT_{0}}\frac{1}{w}\left(\left(\frac{3x^{\theta}}{p^{\theta}}\right)^{w}-\left(\frac{x^{\theta}}{3p^{\theta}}\right)^{w}\right)~{}dw=1+\operatorname{O}\left(\frac{1}{T_{0}}\right).$

Therefore (3.5) becomes

$\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\frac{\alpha p^{\theta}}{U_{\pm}}\
+\operatorname{O}\left(\frac{1}{T_{0}U_{\pm}}\sum_{\begin{subarray}{c}x_{j}<p\leq
x_{j+1}\\\ \sigma_{p}\in C\end{subarray}}\alpha p^{\theta}\right).$ (3.9)

Using $T_{0}=\alpha x^{\theta}$ and collecting the terms from (3.4), (3.8) and
(3.9), we obtain

$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\left(\left[\alpha p^{\theta}\ -\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]-\frac{\alpha
p^{\theta}}{U_{\pm}}\right)$
$\displaystyle\ll\delta\pi_{C}(x_{j},L)^{1/2}\left(1+\frac{x^{(1-\theta)/2}}{(\alpha\omega)^{1/2}}\right)+\frac{\delta}{\alpha
x^{\theta}}\ \pi_{C}(x_{j},L)$ $\displaystyle\quad+\frac{\alpha
x^{\theta}}{T_{1}}\log(\alpha
x^{\theta})\pi_{C}(x_{j},L)+\frac{1}{U_{\pm}}\int_{T_{0}}^{T_{1}}|L(1/2+it)||F(-\theta/2-i\theta
t)|~{}dt$ (3.10)

In order to estimate the integral in the error term above, we use dyadic
division, and bound the following integrals

$\frac{1}{2\pi
i}\int_{1/2+iT^{\prime}}^{1/2+i2T^{\prime}}L(s)H(s)F(-s)~{}ds\ll\frac{1}{U}_{\pm}\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)||F(-\theta/2-i\theta
t)|~{}dt$

for $T_{0}\leq T^{\prime}\leq T_{1}/2$ using Cauchy-Schwarz inequality. This
is achieved by obtaining the following uniform bound for
$T^{\prime}\leq\tau\leq 2T^{\prime}$ and $[x_{j},x_{j+1}]\subseteq[x,2x]$
established in Proposition 4.2 for $K=\mathbb{Q}$

$\displaystyle F(-\theta/2-i\theta\tau)$ $\displaystyle\ll
x^{\theta/2}\left(\log d_{L}+b+\frac{bx\log x}{T^{\prime}}\right)$
$\displaystyle\quad+\frac{|C|}{|G|}x^{(1+\theta)/2}\bigg{(}\log
d_{L}+n_{L}\log T^{\prime}\bigg{)}\left(\frac{\log T^{\prime}}{\log
x}+\frac{x^{1/2}}{T^{\prime}}\right);$

and the mean value estimate below given by Lemma 4.1

$\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)|^{2}~{}dt\ll\alpha{T^{\prime}}+\alpha^{2}x^{\theta}(\log\alpha
x^{\theta}).$

This gives us

$\displaystyle\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)||F(-\theta/2-i\theta
t)|~{}dt$
$\displaystyle\ll\left(\alpha{T^{\prime}}^{2}+\alpha^{2}x^{\theta}T^{\prime}(\log\alpha
x^{\theta})\right)^{1/2}\bigg{\\{}x^{\theta/2}\left(\log d_{L}+b+b\frac{x\log
x}{T^{\prime}}\right)$
$\displaystyle\quad+\frac{|C|}{|G|}x^{(1+\theta)/2}\bigg{(}\log
d_{L}+n_{L}\log T^{\prime}\bigg{)}\left(\frac{\log T^{\prime}}{\log
x}+\frac{x^{1/2}}{T^{\prime}}\right)\bigg{\\}}$ $\displaystyle\ll
x^{\theta/2}\left(\log
d_{L}+b\right)\left(\alpha{T^{\prime}}^{2}+\alpha^{2}T^{\prime}x^{\theta}\log(\alpha
x^{\theta})\right)^{1/2}$ $\displaystyle\quad+bx^{1+\theta/2}\log
x\left(\alpha+\alpha^{2}{T^{\prime}}^{-1}x^{\theta}\log(\alpha
x^{\theta})\right)^{1/2}$
$\displaystyle\quad+\frac{|C|}{|G|}x^{(1+\theta)/2}\left(\log d_{L}+n_{L}\log
T^{\prime}\right)$
$\displaystyle\quad\left(\left(\alpha{T^{\prime}}^{2}+\alpha^{2}T^{\prime}x^{\theta}\log(\alpha
x^{\theta})\right)^{1/2}\frac{\log T^{\prime}}{\log
x}+\left(\alpha+\alpha^{2}{T^{\prime}}^{-1}x^{\theta}\log(\alpha
x^{\theta})\right)^{1/2}x^{1/2}\right).$

Recall that $U_{\pm}\ll\dfrac{\alpha x^{\theta}}{\delta}$, $\alpha
x^{\theta}\leq T^{\prime}<T_{1}$, and $\alpha x^{\theta}\log(\alpha
x^{\theta})\ll{\delta}T_{1}$. Therefore,

$\displaystyle\frac{1}{U_{\pm}}\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)||F(-\theta/2-i\theta
t)|~{}dt$ $\displaystyle\ll\frac{\delta}{(\alpha x^{\theta})^{1/2}}\
\bigg{\\{}\frac{|C|}{|G|}x^{1/2}(\log d_{L}+n_{L}\log
T_{1})\left(T_{1}\frac{\log T_{1}}{\log x}+x^{1/2}\right)$
$\displaystyle\quad+T_{1}\left(\log d_{L}+b\right)+bx\log x\bigg{\\}}.$

We now set

$\displaystyle T_{1}$
$\displaystyle=\frac{\alpha^{3/4}}{(n_{L}\delta\omega)^{1/2}\log
x}x^{(1+3\theta)/4}.$ (3.11)

Note that $\log T_{1}\ll\log x$ since we have assumed
$\displaystyle{\alpha^{1/4}(\omega n_{L}/\delta)^{1/2}(\log x)^{2}\ll
x^{(1-\theta)/4}}.$ Thus, using $\log d_{L}\ll n_{L}\log n_{L}$ (which follows
from the discussion in Serre[Ser72, Section I.3]), we conclude

$\displaystyle\frac{1}{U_{\pm}}\int_{T^{\prime}}^{2T^{\prime}}$
$\displaystyle|L(1/2+it)||F(-\theta/2-i\theta t)|~{}dt\ll(\delta
n_{L}/\omega)^{1/2}\alpha^{1/4}x^{(1+\theta)/4}$
$\displaystyle\quad+\frac{|C|}{|G|}\left((\delta
n_{L}/\omega)^{1/2}\alpha^{1/4}x^{3/4+\theta/4}+\frac{\delta
n_{L}}{\alpha^{1/2}}x^{1-\theta/2}\log x\right).$

Given the value of $T_{1}$, since we use dyadic division of the interval
$[T_{0},T_{1}]$, the number of integrals that we need to add is
$\operatorname{O}(\log x)$. With this, inserting the above estimate in (3.1)
gives us

$\displaystyle\sum_{\begin{subarray}{c}x_{j}<p\leq x_{j+1}\\\ \sigma_{p}\in
C\end{subarray}}\left(\left[\alpha p^{\theta}\ -\delta_{1}\right]-\left[\alpha
p^{\theta}\left(1-\frac{1}{U_{\pm}}\right)-\delta_{1}\right]-\frac{\alpha
p^{\theta}}{U_{\pm}}\right)$
$\displaystyle\ll\delta\pi_{C}(x_{j},L)^{1/2}\left(1+\frac{x^{(1-\theta)/2}}{(\alpha\omega)^{1/2}}\right)+\frac{\delta}{\alpha
x^{\theta}}\ \pi_{C}(x_{j},L)$ (3.12) $\displaystyle\quad+\frac{|C|}{|G|}\log
x\left((\delta n_{L}/\omega)^{1/2}\alpha^{1/4}x^{(3+\theta)/4}+\frac{\delta
n_{L}}{\alpha^{1/2}}x^{1-\theta/2}\log x\right)$ (3.13)
$\displaystyle\quad+(\delta
n_{L}/\omega)^{1/2}\alpha^{1/4}x^{(1+\theta)/4}\log x.$

Note that the error terms in (3.12) can be absorbed into (3.13). Invoking
(3.1), this completes the proof of Theorem 1.1. ∎

###  3.2. Proof of Theorem 1.3

Since the proof is quite similar to that of Theorem 1.1, we point out only the
main differences below and omit the details.

###### Proof.

We proceed as in the proof of Theorem 1.1 with

$\delta_{1}=0,\;\quad\delta_{2}=p^{-\lambda}.$

Note that in this case $\delta=p^{-\lambda}\leq x_{j}^{-\lambda}$ and hence we
can eliminate $\delta$ from the error terms. We follow the proof above until
(3.11) and choose

$T_{1}=\alpha\omega x_{j}^{\theta+\lambda}\log x.$

Following the reasoning after (3.11) in the proof of Theorem 1.1 with the
above value of $T_{1}$, we obtain the asymptotics when $x_{j}<p\leq x_{j+1}$.
Lastly, summing over $j$ gives the desired result claimed in Theorem 1.3. ∎

###  3.3. Proof of Corollary 2

We show

$\\#\left\\{x<p\leq 2x\;:\;\sigma_{p}\in
C_{0}\;\text{and}\;\left\\{\frac{2\sqrt{p}}{\ell}\right\\}\in\left[0,\frac{1}{\ell}\right)\right\\}=\frac{x}{\ell^{2}\log
x}+\operatorname{O}\left(\frac{x^{7/8}}{\ell^{1/4}}\log x\right),$

where $C_{0}$ denotes the union of conjugacy classes of trace zero in
$\mbox{Gal}(L/\mathbb{Q})=\mbox{GL}_{2}(\mathbb{F}_{\ell})$.

Before giving the proof, we first fix some notations. For a group $G$ and
$C\subset G$, let $\delta_{C}:G\to\\{0,1\\}$ denote the class function such
that $\delta_{C}(g)=1$ if and only if $g\in C$. Then,

$\pi_{C}(x,L)=\sum_{\begin{subarray}{c}p\text{ prime }\\\ p\text{ unramified
in }L\end{subarray}\\\ {x<p\leq 2x}}\delta_{C}(\sigma_{p}).$

Let

$\Phi_{C,[\delta_{1},\delta_{2})}(x,L,\alpha):=\sum_{\begin{subarray}{c}p\text{
prime }\\\ p\text{ unramified in }L\\\ {x<p\leq 2x}\\\ \delta_{1}\leq\\{\alpha
p^{\theta}\\}<\delta_{2}\end{subarray}}\delta_{C}(\sigma_{p}).$

We now define an analogue of these functions that include contributions from
ramified primes as well. Let $D_{\mathfrak{p}}$ and $I_{\mathfrak{p}}$ denote
the decomposition and inertia subgroups of $G$, respectively at a chosen prime
ideal $\mathfrak{p}$ lying above $p$. Consider $\mbox{Frob}_{\mathfrak{p}}\in
D_{\mathfrak{p}}/I_{\mathfrak{p}}$, the Frobenius element at $\mathfrak{p}$.
Then, for each integer $m\geq 1$, we define

$\delta_{C}(\sigma_{p}^{m}):=\frac{1}{|I_{\mathfrak{p}}|}\sum_{\begin{subarray}{c}{g\in
D_{\mathfrak{p}}}\\\ {gI_{\mathfrak{p}}=\mbox{Frob}_{\mathfrak{p}}^{m}\in
D_{\mathfrak{p}}/I_{\mathfrak{p}}}\end{subarray}}\delta_{C}(g).$

Note that $\delta_{C}(\sigma_{p}^{m})$ is independent of the choice of
$\mathfrak{p}$ and the above definition agrees with the usual definition of
$\delta_{C}(\sigma_{p}^{m})$ for primes $p$ that are unramified in $L$. Define

$\tilde{\pi}_{C}(x,L):=\sum_{\begin{subarray}{c}{p\text{ prime},\,m\geq 1}\\\
{x<p^{m}\leq
2x}\end{subarray}}\frac{\delta_{C}(\sigma_{p}^{m})}{m}\quad\text{and}\quad\tilde{\Phi}_{C,[\delta_{1},\delta_{2})}(x,L,\alpha):=\sum_{\begin{subarray}{c}{p\text{
prime},\,m\geq 1}\\\ {x<p^{m}\leq 2x}\\\ \delta_{1}\leq\\{\alpha
p^{m\theta}\\}<\delta_{2}\end{subarray}}\frac{\delta_{C}(\sigma_{p}^{m})}{m}.$

With these notations, we state two lemmas from [Zyw15] to be used later, and
we state them for our case when the base field is $\mathbb{Q}$.

######  Lemma 3.1.

[Zyw15, Lemma 2.7] For any subset $C$ of $G$ stable under conjugation,

$\tilde{\pi}_{C}(x,L)=\pi_{C}(x,L)+\operatorname{O}\left(\frac{x^{1/2}}{\log
x}+\log d_{L}\right).$ (3.14)

The following result follows from Proposition 8 of [Ser72].

######  Lemma 3.2.

[Zyw15, Lemma 2.6 (ii)] Let $N$ be a normal subgroup of $G$ and let $C$ be a
subset of $G$ stable under conjugation that satisfies $NC\subseteq C$. Then

$\tilde{\pi}_{C}(x,L)=\tilde{\pi}_{C^{\prime}}(x,L^{N}),$

where $C^{\prime}$ is the image of $C$ in $G/N=\mbox{Gal}(L^{N}/\mathbb{Q})$.

We are now ready to prove Corollary 2.

###### Proof.

Consider the extension $L/\mathbb{Q}$ with $L=\mathbb{Q}(E[\ell])$. Observe
that $C_{0}$ is stable under multiplication by $H_{\ell}$, the subgroup of
scalar matrices in $\mbox{GL}_{2}(\mathbb{F}_{\ell})$. Moreover, it is the
inverse image of $C_{0}^{\prime}$, the subset of order two elements in
$G_{\ell}^{\prime}:=G_{\ell}/H_{\ell}=\mbox{PGL}_{2}(\mathbb{F}_{\ell})$.
Applying Lemma 2, we obtain

$\tilde{\pi}_{C_{0}}(x,L)=\tilde{\pi}_{C_{0}^{\prime}}(x,L^{H_{\ell}}).$

Therefore, using (3.14)

$\displaystyle\\#\left\\{x<p\leq 2x:\sigma_{p}\in C_{0},\;p\text{ unramified
and}\;\left\\{\frac{2\sqrt{p}}{\ell}\right\\}\in\left[0,\frac{1}{\ell}\right)\right\\}$
$\displaystyle=\Phi_{C_{0},[0,\frac{1}{\ell})}\left(x,L,\frac{2}{\ell}\right)$
$\displaystyle=\tilde{\Phi}_{C_{0},[0,\frac{1}{\ell})}\left(x,L,\frac{2}{\ell}\right)+\operatorname{O}\left(\frac{x^{1/2}}{\log
x}+\log d_{L}\right)$
$\displaystyle=\tilde{\Phi}_{C_{0}^{\prime},[0,\frac{1}{\ell})}\left(x,L^{H_{\ell}},\frac{2}{\ell}\right)+\operatorname{O}\left(\frac{x^{1/2}}{\log
x}+\log d_{L}\right)$
$\displaystyle=\Phi_{C_{0}^{\prime},[0,\frac{1}{\ell})}\left(x,L^{H_{\ell}},\frac{2}{\ell}\right)+\operatorname{O}\left(\frac{x^{1/2}}{\log
x}+\log d_{L}\right).$

Using Theorem 1.1 and Chebotarev density theorem to the sub-extension
$L^{H_{\ell}}/\mathbb{Q}$,

$\displaystyle\Phi_{C_{0}^{\prime},[0,\frac{1}{\ell})}\left(x,L^{H_{\ell}},\frac{2}{\ell}\right)+\operatorname{O}\left(\frac{x^{1/2}}{\log
x}+\log d_{L}\right)$
$\displaystyle=\frac{\pi_{C_{0}^{\prime}}(x,L^{H_{\ell}})}{\ell}+\operatorname{O}\left(\frac{x}{\ell^{2}\omega\log
x}+\omega^{1/2}\frac{x^{7/8}}{\ell^{1/4}}\log x+\omega\ell^{3/2}x^{3/4}(\log
x)^{2}\right)$
$\displaystyle=\frac{\pi(x)}{\ell^{2}}+\operatorname{O}\left(\frac{x}{\ell^{2}\omega\log
x}+\omega^{1/2}\frac{x^{7/8}}{\ell^{1/4}}\log x+\omega\ell^{3/2}x^{3/4}(\log
x)^{2}\right)$

noting that $\log d_{L}\ll[L^{H_{\ell}}:\mathbb{Q}]\ll\ell^{3}$ and $\ell\ll
x^{1/14}\omega^{-2/7}\log^{-8/7}x$. This completes the proof. ∎

##  4\. Proofs of intermediate results

###  4.1. Mean value estimation of $L(1/2+it)$

######  Lemma 4.1.

$\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)|^{2}~{}dt\ll\alpha
T^{\prime}+{\alpha}^{2}x^{\theta}\log(\alpha x^{\theta}).$

###### Proof.

We have

$\displaystyle\int_{T^{\prime}}^{2T^{\prime}}\left|L(1/2+it)\right|^{2}~{}dt=\alpha\int_{T^{\prime}}^{2T^{\prime}}\left|\sum_{\alpha
x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}(m+\delta_{1})^{-1/2-it}\right|^{2}~{}dt$
$\displaystyle=\alpha\int_{T^{\prime}}^{2T^{\prime}}\sum_{\alpha
x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}(m+\delta_{1})^{-1}+\sum_{\alpha
x^{\theta}/3-\delta_{1}<k\neq m\leq 3\alpha
x^{\theta}-\delta_{1}}\frac{(k+\delta_{1})^{-1/2+it}}{(m+\delta_{1})^{1/2+it}}~{}dt$
$\displaystyle=\sum_{\alpha x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}\frac{\alpha
T^{\prime}}{(m+\delta_{1})}+\operatorname{O}\left(\sum_{\alpha
x^{\theta}/3-\delta_{1}<k<m\leq 3\alpha
x^{\theta}-\delta_{1}}\frac{\alpha((m+\delta_{1})(k+\delta_{1}))^{-1/2}}{\log((m+\delta_{1})/(k+\delta_{1}))}\right).$

Using $\delta_{1}<\alpha x^{\theta}/3$ and rewriting the sum in the error
term, we obtain

$\displaystyle\sum_{\alpha x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}\sum_{r=1}^{(m+\delta_{1})/9}\frac{\alpha}{(m+\delta_{1})\sqrt{1-r/(m+\delta_{1}})\log\left(1-r/(m+\delta_{1})\right)}$
$\displaystyle\ll\sum_{\alpha x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}\sum_{r=1}^{(m+\delta_{1})/9}\frac{\alpha}{r}\ll\alpha^{2}x^{\theta}\log(\alpha
x^{\theta}).$

This gives us

$\displaystyle\int_{T^{\prime}}^{2T^{\prime}}|L(1/2+it)|^{2}dt$
$\displaystyle=$ $\displaystyle\alpha T^{\prime}\sum_{\alpha
x^{\theta}/3-\delta_{1}<m\leq 3\alpha
x^{\theta}-\delta_{1}}(m+\delta_{1})^{-1}+\operatorname{O}\left(\alpha^{2}x^{\theta}\log(\alpha
x^{\theta})\right)$ $\displaystyle\ll$ $\displaystyle\alpha
T^{\prime}+\alpha^{2}x^{\theta}\log(\alpha x^{\theta}).$

This completes the proof of the lemma. ∎

###  4.2. Estimation of $F(-\theta/2-i\theta t)$

For each sub-interval $[x_{j},x_{j+1}]\subset[x,2x]$ we prove the following.

######  Proposition 4.2.

Let $\theta\in[0,1]$ be fixed. Given a Galois extension of number fields $L$
over $K$, and $C$ being a union of $b$ conjugacy class in Gal$(L/K)$, define

$F(-\theta/2-i\theta\tau)=\sum\limits_{\begin{subarray}{c}x_{j}<N(\mathfrak{p})\leq
x_{j+1}\\\ \sigma_{\mathfrak{p}}\in
C\end{subarray}}(N\mathfrak{p})^{\theta/2+i\theta\tau}.$

Assume the Riemann Hypothesis for the Dedekind zeta function of the field
extension $L/K$. Then

$\displaystyle F(-\theta/2-i\theta\tau)$
$\displaystyle\ll\frac{|C|}{|G|}x^{(1+\theta)/2}(\log d_{L}+n_{L}\log
T^{\prime})\left(\frac{\log T^{\prime}}{\log
x}+\frac{x^{1/2}}{T^{\prime}}\right)$
$\displaystyle\quad+x^{\theta/2}\left(\log d_{L}+bn_{K}+bn_{K}\frac{x\log
x}{T^{\prime}}\right)$

uniformly for $0<T^{\prime}\leq\tau\leq 2T^{\prime}$.

The proof follows along the same lines as in [LO77]. The function
$F(-\theta/2-i\theta\tau)$ here is similar to that of $\pi_{C}(x,L/K)$ in
[LO77], the main difference being a shift in the complex variable
$s=\sigma+it$ by $\theta/2+i\theta\tau$. While the shift in the real part by
$\theta/2$ results in a factor of $x^{\theta/2}$ tagging along with the error
terms obtained in [LO77], the shift in the imaginary part is where the saving
is obtained. To be precise, we choose a contour that is a box which avoids the
real axis, therefore the only poles in the interior are the non-trivial zeros
of $L(s,\chi,L/E)$, and a pole at $-\theta/2-i\theta\tau$. In particular, the
residue from the pole at $s=1$ that makes up the main term in the proof by
Lagarias-Odlyzko [LO77] does not appear here, giving us a power saving under
GRH.

We now provide details of the proof.

###### Proof.

We first consider the function

$\Psi_{C}(-\theta/2-i\theta\tau):=\sum\limits_{\begin{subarray}{c}N\mathfrak{p}^{m}\in(x_{j},x_{j+1}]\\\
\mathfrak{p}\text{:unramified}\\\
\left[\frac{L/K}{\mathfrak{p}}\right]^{m}=C\end{subarray}}\frac{\log
N\mathfrak{p}}{N\mathfrak{p}^{m(-\theta/2-i\theta\tau)}}$

for a single conjugacy class $C$ where $\Lambda$ denotes the von-Mangoldt
function. We use partial summation to pass on to the bounds for
$F(-\theta/2-i\theta\tau)$. As in [LO77], in order to use Hecke $L$-functions,
we need to consider the ramified primes as well, (which are later removed).
For $\Re s>1$, let

$Z(s):=-\frac{|C|}{|G|}\sum_{\chi}\bar{\chi}(g)\frac{L^{\prime}}{L}(s,\chi,L/E),$
(4.1)

where $\chi$ runs over the irreducible characters of $H=Gal(L/E)$ and $E$ is
the fixed field of $H$, the cyclic subgroup of $G$ generated by a chosen
element $g\in C$. Note that

$Z(s)=\sum_{\mathfrak{p},m}\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})(N\mathfrak{p})^{-ms}$

where for an unramified prime $\mathfrak{p}\subseteq\mathcal{O}_{K},$

$\Theta(\mathfrak{p}^{m})=\begin{cases}1&\text{if
}\left[\frac{L/K}{\mathfrak{p}}\right]^{m}=C\\\ 0&\text{otherwise}\end{cases}$

and $|\Theta(\mathfrak{p}^{m})|\leq 1$ if $\mathfrak{p}$ ramifies in $L$. We
use Perron’s formula to estimate $F(-\theta/2-i\theta\tau)$ by considering
partial sums of $Z(s)$, which include the ramified primes as well. Define

$I(x_{j},T):=I_{C}(x_{j},T,\theta,\tau)=\frac{1}{2\pi
i}\int\limits_{\sigma_{0}-iT}^{\sigma_{0}+iT}Z(s-\theta/2-i\theta\tau)\frac{{x_{j+1}}^{s}-{x_{j}}^{s}}{s}ds$
(4.2)

where $T=\theta T^{\prime}/2$ and $\sigma_{0}=1+\theta/2+1/\log x$. Then,

$\displaystyle|\Psi_{C}(-\theta/2-i\theta\tau)-I(x_{j},T)|$
$\displaystyle\leq\left|I(x_{j},T)-\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
x_{j}<N\mathfrak{p}\leq
x_{j+1}\end{subarray}}\frac{\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})}{(N\mathfrak{p})^{m(-\theta/2-i\theta\tau)}}\right|$
$\displaystyle\quad+\left|\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
x_{j}<N\mathfrak{p}\leq x_{j+1}\\\
\mathfrak{p}\text{:ramified}\end{subarray}}\frac{\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})}{(N\mathfrak{p})^{m(-\theta/2-i\theta\tau)}}\right|.$
(4.3)

The two terms on the right hand side of the above equation are now estimated.
Using Lemma 3.1 of [LO77], we have,

$\displaystyle\left|I(x_{j},T)-\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
x_{j}<N\mathfrak{p}\leq
x_{j+1}\end{subarray}}\frac{\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})}{(N\mathfrak{p})^{m(-\theta/2-i\theta\tau)}}\right|$
$\displaystyle\hskip
28.45274pt\leq\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
N\mathfrak{p}^{m}=x_{j+1}\text{ or
}N\mathfrak{p}^{m}=x_{j}\end{subarray}}\left(\frac{\log
N\mathfrak{p}}{(N\mathfrak{p}^{m})^{-\theta/2}}+\frac{\sigma_{0}}{T}\right)$
$\displaystyle\hskip
49.79231pt+\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
N\mathfrak{p}^{m}\neq
x_{j+1}\end{subarray}}\left(\frac{x_{j+1}}{N\mathfrak{p}^{m}}\right)^{\sigma_{0}}\min\left(1,T^{-1}\left|\log\frac{x_{j+1}}{N\mathfrak{p}^{m}}\right|^{-1}\right)\frac{\log
N\mathfrak{p}}{(N\mathfrak{p}^{m})^{-\theta/2}}$ $\displaystyle\hskip
49.79231pt+\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
N\mathfrak{p}^{m}\neq
x_{j}\end{subarray}}\left(\frac{x_{j}}{N\mathfrak{p}^{m}}\right)^{\sigma_{0}}\min\left(1,T^{-1}\left|\log\frac{x_{j}}{N\mathfrak{p}^{m}}\right|^{-1}\right)\frac{\log
N\mathfrak{p}}{(N\mathfrak{p}^{m})^{-\theta/2}}.$

Following arguments from [LO77] to estimate the terms on the right side of the
above inequality, and noting that $x_{j}\leq 2x$ for each $j=0,\ldots,B$, we
get

$I(x_{j},T)-\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
x_{j}<N\mathfrak{p}\leq
x_{j+1}\end{subarray}}\frac{\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})}{(N\mathfrak{p})^{m(-\theta/2-i\theta\tau)}}\ll
x^{\theta/2}n_{K}\log
x+n_{K}\frac{\sigma_{0}}{T}+n_{K}\frac{x^{1+\theta/2}\log^{2}x}{T}.$ (4.4)

Moreover,

$\displaystyle\sum\limits_{\begin{subarray}{c}\mathfrak{p},m\\\
x_{j}<N\mathfrak{p}\leq x_{j+1}\\\
\mathfrak{p}\text{:ramified}\end{subarray}}\frac{\Theta(\mathfrak{p}^{m})\log(N\mathfrak{p})}{(N\mathfrak{p})^{m(-\theta/2-i\theta\tau)}}\ll
x^{\theta/2}\log x\log d_{L}.$ (4.5)

Putting (4.4) and (4.5) together in (4.2), we see that

$\Psi_{C}(-\theta/2-i\theta\tau)=I(x_{j},T)+\operatorname{O}\left(x^{\theta/2}\log
x\left(\log d_{L}+n_{K}+\frac{n_{K}x\log x}{T}\right)\right).$ (4.6)

Next, we estimate $I(x_{j},T)$. From (4.2) and (4.1), we have

$I(x_{j},T)=-\frac{|C|}{|G|}\sum_{\chi}\bar{\chi}(g)\frac{1}{2\pi
i}\int\limits_{\sigma_{0}-iT}^{\sigma_{0}+iT}\frac{L^{\prime}}{L}\left(s-{\theta}/{2}-i\theta\tau,\chi,L/E\right)\frac{x_{j+1}^{s}-x_{j}^{s}}{s}~{}ds,$

where $\chi$ runs through irreducible characters of $H=\langle g\rangle$,
$T^{\prime}\leq\tau\leq 2T^{\prime}$ is fixed, and $T=\theta T^{\prime}/2$. We
make the change of variable $s\leftrightarrow s-\frac{\theta}{2}-i\theta\tau$
to rewrite

$I(x_{j},T)=-\frac{|C|}{|G|}\sum_{\chi}\frac{\bar{\chi}(g)}{2\pi
i}\int\limits_{1+\frac{1}{\log x}-iT-i\theta\tau}^{1+\frac{1}{\log
x}+iT-i\theta\tau}\frac{L^{\prime}}{L}\left(s,\chi,L/E\right)\frac{x_{j+1}^{s+\theta/2+i\theta\tau}-x_{j}^{s+\theta/2+i\theta\tau}}{s+\theta/2+i\theta\tau}~{}ds.$

Abbreviating $\frac{L^{\prime}}{L}(s,\chi,L/E)$ by
$\frac{L^{\prime}}{L}(s,\chi)$, we evaluate for each character $\chi$ of $H$,
the integral

$I_{\chi}(x_{j},T):=\frac{1}{2\pi i}\int\limits_{1+\frac{1}{\log
x}-iT-i\theta\tau}^{1+\frac{1}{\log
x}+iT-i\theta\tau}\frac{L^{\prime}}{L}(s,\chi)\frac{x_{j}^{s+\theta/2+i\theta\tau}}{s+\theta/2+i\theta\tau}~{}ds$

for each $j=0,1,\ldots,B$. We may assume that $T+\theta\tau$ and
$T-\theta\tau$ don’t coincide with the imaginary part of a zero of any of the
$L(s,\chi)$. To estimate this integral, we move the line of integration and
consider the integral over a rectangle and apply Cauchy’s theorem. More
specifically, for $J:=m+\frac{1}{2}$ where $m$ is a non-negative integer, let
$B_{T,J,\theta}$ be the positively oriented rectangle with vertices at
$1+\frac{1}{\log x}-i(T+\theta\tau),\,1+\frac{1}{\log
x}+i(T-\theta\tau),\,-J-\frac{\theta}{2}+i(T-\theta\tau)$ and
$-J-\frac{\theta}{2}-i(T+\theta\tau).$ Observe that this box does not
intersect the real-axis, because $T-\theta\tau<0$ for all
$\tau\in[T^{\prime},2T^{\prime}]$. Define

$I_{\chi}(x_{j},T,J):=\frac{x_{j}^{\theta/2+i\theta\tau}}{2\pi
i}\int_{B_{T,J,\theta}}\frac{L^{\prime}}{L}(s,\chi)\frac{x_{j}^{s}}{s+\theta/2+i\theta\tau}~{}ds.$

Now we estimate the error term

$\displaystyle R_{\chi}(x_{j},T,J):=I_{\chi}(x_{j},T,J)-I_{\chi}(x_{j},T)$
(4.7)

uniformly for each $j=0,\ldots,B$. Here, the error $R_{\chi}(x_{j},T,J)$
consists of sum of one vertical integral $V_{\chi}(x_{j},T,J)$, and two
horizontal integrals $H_{\chi}(x_{j},T,J)$ and $H^{*}_{\chi}(x_{j},T)$ which
we now estimate, following the line of proof in [LO77, Section 6, Lemma 6.2].
We deduce

$\displaystyle V_{\chi}(x_{j},T,J)$ $\displaystyle:=\frac{1}{2\pi
i}\int\limits_{T}^{-T}\frac{x_{j}^{-J+it}}{-J+it}\frac{L^{\prime}}{L}(-J-\theta/2+i(t-\theta\tau),\chi)~{}dt$
$\displaystyle\ll\frac{x^{-J}}{J}T\left(\log
A(\chi)+n_{E}\log(|T+\theta\tau|+|J+\theta/2|\right);$ $\displaystyle
H_{\chi}(x_{j},T,J)$ $\displaystyle:=\frac{x_{j}^{\theta/2+i\theta\tau}}{2\pi
i}\int\limits_{-J-\theta/2}^{-1/4}\frac{x_{j}^{\sigma-
iT}}{\sigma+\theta/2-iT}\frac{L^{\prime}}{L}(\sigma-i(T+\theta\tau),\chi)~{}d\sigma$
$\displaystyle\quad-\frac{x_{j}^{\theta/2+i\theta\tau}}{2\pi
i}\int\limits_{-J-\theta/2}^{-1/4}\frac{x_{j}^{\sigma+iT}}{\sigma+\theta/2+iT}\frac{L^{\prime}}{L}(\sigma+i(T-\theta\tau),\chi)~{}d\sigma$
$\displaystyle\ll\frac{x^{-1/4+\theta/2}}{T\log x}\left(\log
A(\chi)+n_{E}\log|T+\theta\tau|\right);$

and lastly

$\displaystyle H^{*}_{\chi}(x_{j},T)$
$\displaystyle:=\frac{x_{j}^{\theta/2+i\theta\tau}}{2\pi
i}\int\limits_{-1/4}^{1+1/\log x}\frac{x_{j}^{\sigma-
iT}}{\sigma+\theta/2-iT}\frac{L^{\prime}}{L}(\sigma-i(T+\theta\tau),\chi)~{}d\sigma$
$\displaystyle\quad-\frac{x_{j}^{\theta/2+i\theta\tau}}{2\pi
i}\int\limits_{-1/4}^{1+1/\log
x}\frac{x_{j}^{\sigma+iT}}{\sigma+\theta/2+iT}\frac{L^{\prime}}{L}(\sigma+i(T-\theta\tau),\chi)~{}d\sigma$
$\displaystyle=\frac{1}{2\pi i}\int_{-1/4}^{1+1/\log
x}\frac{x_{j}^{\sigma+\theta/2-iT}}{\sigma+\theta/2-iT}\sum_{\begin{subarray}{c}\rho\\\
|\gamma-(T+\theta\tau)|\leq
1\end{subarray}}\frac{~{}d\sigma}{\sigma-i(T+\theta\tau)-\rho}$
$\displaystyle\quad-\frac{1}{2\pi i}\int_{-1/4}^{1+1/\log
x}\frac{x_{j}^{\sigma+\theta/2+iT}}{\sigma+\theta/2+iT}\sum_{\begin{subarray}{c}\rho\\\
|\gamma+(T-\theta\tau)|\leq
1\end{subarray}}\frac{d\sigma}{\sigma+i(T-\theta\tau)-\rho}$
$\displaystyle\quad+\operatorname{O}\left(\frac{x^{1+\theta/2}}{T\log x}(\log
A(\chi)+n_{E}\log|T+\theta\tau|)\right),$

where the sum above is taken over the non-real zeros $\rho$ of $L(s,\chi)$ and
$A(\chi)=d_{E}N_{E/\mathbb{Q}}(f_{\chi})$, $f_{\chi}$ being the conductor of
$\chi$. The proof of Lemma $6.3$ in [LO77] can be modified to show that

$\displaystyle\int\limits_{-1/4}^{1+1/\log
x}\frac{x_{j}^{\sigma+\theta/2-iT}}{\sigma+\theta/2-iT}\sum_{\begin{subarray}{c}\rho\\\
|\gamma-(T+\theta\tau)|\leq
1\end{subarray}}\frac{1}{\sigma-i(T+\theta\tau)-\rho}~{}d\sigma$
$\displaystyle\hskip 28.45274pt\ll\frac{x^{1+1/\log x}}{T\log
x}n_{\chi}(T+\theta\tau)$ $\displaystyle\hskip
28.45274pt\ll\frac{x^{1+\theta/2}\log x}{T}(\log
A(\chi)+n_{E}\log|T+\theta\tau|).$

Here $n_{\chi}(t)$ denotes the number of zeros $\rho=\beta+i\gamma$ of
$L(s,\chi)$ with $0<\beta<1$ and $|\gamma-t|\leq 1$. A similar estimate holds
for the sum over $\rho$ with $|\gamma+(T-\theta\tau)|\leq 1$. Therefore, for
each $j=0,1,\ldots,B$,

$H_{\chi}^{*}(x_{j},T)\ll\frac{x^{1+\theta/2}\log x}{T}\left(\log
A(\chi)+n_{E}\log|T+\theta\tau|\right).$

Note that the estimate for $H_{\chi}(x_{j},T)$ is bounded above by the
estimate for $H_{\chi}^{*}(x_{j},T)$. Therefore, from (4.7),

$\displaystyle R_{\chi}(x_{j},T,J)$ $\displaystyle\ll\frac{x^{1+\theta/2}\log
x}{T}\left(\log A(\chi)+n_{E}\log|T+\theta\tau|\right)$
$\displaystyle+\frac{x^{-J}}{J}T\left(\log
A(\chi)+n_{E}\log(|T-\theta\tau|+|J+\theta/2|\right).$ (4.8)

We remark here that one only needs to consider the first term above; the
second term goes to zero as $J\to\infty$. Next, by Cauchy’s theorem,
$I_{\chi}(x_{j},T,J)$ is the sum of the residues at the poles of the integrand
inside $B_{t,J}$. For our specified contour, the poles occur only at the non-
real zeros of $L(s,\chi)$, and at $s=-\theta/2-i\theta\tau$. This gives

$I_{\chi}(x_{j},T,J)=\sum\limits_{|\gamma+\theta\tau|<T}\frac{x_{j}^{\rho+\theta/2+i\theta\tau}}{\rho+\theta/2+i\theta\tau}+\frac{L^{\prime}}{L}(-\theta/2-i\theta\tau).$
(4.9)

The term $\frac{L^{\prime}}{L}(-\theta/2-i\theta\tau)$ is estimated using the
following lemma which is a slightly general version of Lemma 6.2 in [LO77] and
can be proved essentially using the same arguments, so we omit the details
here.

######  Lemma 4.3.

If $s=\sigma+it$ with $\sigma\leq-\theta/2$ and $|s+m|\geq\theta/2$ for all
non-negative integers $m$, then

$\frac{L^{\prime}}{L}(s,\chi)\ll\log A(\chi)+n_{E}\log(|s|+2).$

Applying these bounds, we get

$\frac{L^{\prime}}{L}(-\theta/2-i\theta\tau)\ll\log
A(\chi)+n_{E}\log(|\theta/2+i\theta\tau|+2).$ (4.10)

Using (4.10), (4.9) and (4.2) in (4.7), we have

$\displaystyle I_{\chi}(x_{j},T)$
$\displaystyle=\sum\limits_{|\gamma+\theta\tau|<T}\frac{x_{j}^{\rho+\theta/2+i\theta\tau}}{\rho+\theta/2+i\theta\tau}$
$\displaystyle\quad+\operatorname{O}\left(\left(\frac{x^{1+\theta/2}\log
x}{T}+1\right)\left(\log A(\chi)+n_{E}\log|T+\theta\tau|\right)\right).$

We plug this into the definition of $I(x_{j},T)$ to obtain

$\displaystyle I(x_{j},T)-S(x_{j+1},T)+S(x_{j},T)$
$\displaystyle\ll\frac{|C|}{|G|}\left(\frac{x^{1+\theta/2}\log
x}{T}+1\right)\sum_{\chi}\left(\log A(\chi)+n_{E}\log|T+\theta\tau|\right)$
$\displaystyle\ll\frac{|C|}{|G|}\left(\frac{x^{1+\theta/2}\log
x}{T}+1\right)(\log d_{L}+n_{L}\log|T+\theta\tau|),$ (4.11)

where

$S(y,T):=\frac{|C|}{|G|}\sum_{\chi}\bar{\chi}(g)\sum\limits_{|\gamma+\theta\tau|<T}\frac{y^{\rho+\theta/2+i\theta\tau}}{\rho+\theta/2+i\theta\tau}.$

Under GRH, with slight modification to the calculation in the proof of Theorem
$9.1$ of [LO77], and summing over $\chi$, we get

$\displaystyle S(x_{j},T)\ll\frac{|C|}{|G|}x^{1/2+\theta/2}\log T(\log
d_{L}+n_{L}\log|T+\theta\tau|)$

Finally, using the above bounds in (4.2) and recalling (4.6), we conclude

$\displaystyle\Psi_{C}(-\theta/2-i\theta\tau)$
$\displaystyle\ll\frac{|C|}{|G|}x^{1/2+\theta/2}\left(\log
d_{L}+n_{L}\log|T+\theta\tau|\right)\left(\log T+\frac{x^{1/2}}{T}\log
x\right)$ $\displaystyle\quad+x^{\theta/2}\log x\left(\log
d_{L}+n_{K}+n_{K}\frac{x\log x}{T}\right).$

Using partial summation and $\tau\leq 2T^{\prime}$, and setting $T=\theta
T^{\prime}/2$, one gets the desired bounds for $F$ in the case of a fixed
conjugacy class $C$.

Next, suppose $C=\cup_{m=1}^{b}C_{m}$ is a union of $b$ conjugacy classes for
some integer $b\geq 1$. Then, using the estimates established for each
conjugacy class $C_{m}$ and summing over $m$, we have

$\displaystyle F(-\theta/2-i\theta\tau)$
$\displaystyle\ll\frac{\sum_{m=1}^{b}|C_{j}|}{|G|}x^{(1+\theta)/2}\bigg{(}\log
d_{L}+n_{L}\log T^{\prime}\bigg{)}\left(\frac{\log T^{\prime}}{\log
x}+\frac{x^{1/2}}{T^{\prime}}\right)$
$\displaystyle\quad+x^{\theta/2}\left(\log d_{L}+bn_{K}+bn_{K}\frac{x\log
x}{T^{\prime}}\right)$
$\displaystyle=\frac{|C|}{|G|}x^{(1+\theta)/2}\bigg{(}\log d_{L}+n_{L}\log
T^{\prime}\bigg{)}\left(\frac{\log T^{\prime}}{\log
x}+\frac{x^{1/2}}{T^{\prime}}\right)$
$\displaystyle\quad+x^{\theta/2}\left(\log d_{L}+bn_{K}+bn_{K}\frac{x\log
x}{T^{\prime}}\right),$

noting that the error term in (4.5) that contributes $x^{\theta/2}\log d_{L}$
remains unchanged even if $C$ is a single class or a union of conjugacy
classes. This completes the proof of the proposition. ∎

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