# Suppression of singularities of solutions of the Euler-Poisson system with
density-dependent damping

Olga S. Rozanova<EMAIL_ADDRESS>Department of Mechanics and
Mathematrics, Moscow State University, Moscow 119991 Russia

###### Abstract

We find a sharp condition on the density-dependent coefficient of damping of a
one-dimensional repulsive Euler-Poisson system, which makes it possible to
suppress the formation of singularities in the solution of the Cauchy problem
with arbitrary smooth data. In the context of plasma physics, this means the
possibility of suppressing the breakdown of arbitrary oscillations of cold
plasma.

###### keywords:

Euler-Poisson system , equations of cold plasma , singularity formation ,
density dependent damping

###### MSC:

35F55 , 35Q60 , 35B44 , 35B20 , 35L80

##  1 Introduction

We consider the system of Euler-Poisson equations describing the behavior of
cold plasma for velocity $V$, electron density $n>0$ and electric field
potential $\Psi$ in the following form:

$\dfrac{\partial V}{\partial t}+(V\cdot\nabla)V=-\nabla\Psi-\nu
V,\quad\dfrac{\partial n}{\partial t}+{\rm div}\,(nV)=0,\quad\Delta\Psi=1-n.$
(1)

All components of the solution are assumed to be functions of time $t\geqslant
0$ and point $x\in{\mathbb{R}}^{n}$,
$\int\limits_{{\mathbb{R}}^{n}}(n-1)\,dx=\rm const$, $\nu\geqslant 0$ is the
damping factor.

This system is a pressureless variant of the general Euler-Poisson system,
having numerous physical applications, see [1] for references. One of the
crucial questions is the study of the Cauchy problem and the analysis of the
possibility of the existence of a globally in time smooth solution. The model
without pressure is somewhat simpler from a mathematical point of view, since
it allows one to obtain criteria for the formation of a singularity from the
initial data.

In [1], many versions of the model without pressure, including those with
constant damping and viscosity, with both zero and non-zero backgrounds, have
been studied. In all these cases, it is possible to find the initial data
leading to a blow-up in a finite time. Moreover, this possibility still
remains if pressure and heat diffusion are added to the model [2].

In recent years, the pressureless model has attracted great interest, since it
is very convenient to describe the wake wave in the cold plasma generated by a
laser pulse in order to create a new type of accelerator, [3] and references
therein. It is generally known that the plasma oscillation tends to blow-up,
forming a gradient catastrophe in the velocity component and a delta
singularity in the density component. After the moment of the singularity
formation, the cold plasma model loses its relevance; therefore, the
conditions on the initial data or other parameters that make it possible to
maintain a smooth solution as long as possible or, possibly, guaranteeing a
global in time smooth solution is a key question for all the theory.

In plasma physics, the damping factor $\nu$ corresponds to the frequency of
the electron-ion collisions, this value is very small from the physical point
of view. Depending on the model, the electron-ion collisions either can be
neglected or taken into account. In the recent paper [4], for a very
particular solution in the $1D$ case the authors showed numerically that if
$\nu=\nu_{0}n$, where $\nu_{0}$ is a positive constant, than the oscillations
never blow up.

Our main question is whether this result is valid for all possible initial
data and can it be substantiated analytically? In addition, if $\nu$ is a
smooth function of $n$, what conditions must be imposed on $\nu(n)$ to ensure
the global in time smoothness of the solution to the Cauchy problem for any
given data?

In this paper, we focus on the $1D$ case, since explicit analytical results
can be obtained here, and we rewrite (1) in a form more accepted in plasma
physics:

$\dfrac{\partial V}{\partial t}+V\dfrac{\partial V}{\partial
x}=-E-\nu(n)V,\quad\dfrac{\partial E}{\partial t}+V\dfrac{\partial E}{\partial
x}=V,\quad n=1-\dfrac{\partial E}{\partial x},$ (2)

see [3] for details. Here $E=\nabla\Psi$ is the vector of electric field.
System (2) will be considered together with the Cauchy data

$(V,E)|_{t=0}=(V_{0}(x),E_{0}(x))\in{\mathcal{A}}({\mathbb{R}}).$ (3)

For $\nu\neq\rm const$ system (2), do not belong to the symmetric hyperbolic
type, therefore we cannot guarantee that the solution to the Cauchy problem a
local solution as smooth as initial data in the Sobolev norm. Therefore we
have to prescribe the analyticity to initial data to use the Cauchy-
Kovalevskaya theorem and show that problem (2), (3) has a local in time unique
analytical solution.

Problem (2), (3) was completely analyzed for $\nu=0$ in [5] and $\nu=\rm
const>0$ in [6], where sharp conditions on initial data to guaranty a globally
in time smooth solution were found (see the analogous result for another
context in [1]). It was found that even for an arbitrarily large constant
frequency of collisions there exist data implying a finite time singularity
formation.

In the present work, we show that by choosing an appropriate density-dependent
damping factor one can obtain a globally smooth solution for any smooth Cauchy
data, i.e. completely remove the singularity formation. Namely, for our
prototypic function $\nu(n)=\nu_{0}n^{\gamma}$ the threshold value is
$\gamma=1$. For $\gamma\geqslant 1$ the solution to (2), (3) does not form the
gradient catastrophe.

The paper is organized as follows. In Section 2, we consider special
solutions, linear with respect to the spatial variable (the so-called affine
solutions) and prove the exact condition for eliminating blow-up. In Section 3
we prove a similar result for arbitrary initial data. Section 4 discusses
issues related to this problem.

##  2 Affine solutions

First, we consider a special form of solutions:

$V=a(t)x+A(t),\quad E=b(t)x+B(t).$ (4)

######  Theorem 2.1.

Let $f(n)\in{\mathcal{A}}({\mathbb{R}}_{+})$ be a nonnegative function
satisfying condition

$\int\limits_{\eta_{0}>0}^{+\infty}\dfrac{f(\eta)}{\eta^{2}}\,d\eta=\infty.$
(5)

It $\nu(n)=\epsilon f(n)$, $\epsilon={\rm const}>0$, then derivatives of the
solution to problem (2), (3), with data (4) are bounded in time for any choice
of the data. Otherwise, one can find the data such that the derivatives of
solution blow up in a finite time.

###### Proof.

We substitute the ansatz (4) in (1) to obtain

$\displaystyle\dot{a}=-a^{2}-b-\epsilon f(1-b)a,$
$\displaystyle\quad\dot{b}=(1-b)a,$ (6) $\displaystyle\dot{A}=-A(a-\epsilon
f(1-b))-B,$ $\displaystyle\quad\dot{B}=(1-b)A,$ (7)

Since $n=1-b>0$, we consider the domain $b<1$.

The couple of equations (6) splits off from the system, and the second couple
(7) is linear with respect to $A,B$ with the coefficients found in the
previous step. Therefore, if we want to study conditions for a blow-up of the
solution (4), it is sufficient to consider the behavior of the phase curve of
the autonomous system (6), given as

$\displaystyle\dfrac{da}{db}=-\frac{a^{2}+b}{(1-b)a}-\epsilon\frac{f(1-b)}{1-b}.$
(8)

For an arbitrary $f$ equation (8) cannot be integrated explicitly, however, it
can be considered as a regular perturbation of (8) at $\epsilon=0$, which
solution is

$\displaystyle a=\pm\sqrt{1-2b+C(1-b)^{2}}.$ (9)

Here the constant $C=\dfrac{a_{0}^{2}+2b_{0}-1}{(1-b_{0})^{2}}$, with
$a_{0}=a(0)$, $b_{0}=b(0)$.

The analysis of the phase plane shows that a point on the phase plane moves
from the upper half-plane $a>0$ to the lower half-plane $a<0$ and there can
come back to the upper half-plane or go to minus-infinity. The latter
signifies the blowup of derivatives of the solution. If $C<0$, the curve on
the phase plane is bounded (it is an ellipse), otherwise $a(t)$ and $b(t)$
move along a parabola ($C=0$) or hyperbola ($C>0$) and therefore go to minus-
infinity within a finite time (see [5] for details).

Our main question is whether correctors due to parameter $\epsilon$ can change
the behavior of trajectory going to infinity and turn it to the upper half-
plane $a>0$.

When analyzing the phase portrait of the perturbed system, we point out the
following elementary facts, illustrated in Figure 1.

1\. If the initial point of a phase curve is situated in the upper half-plane
$a>0$, within a finite time any point $(b,a)$ turns out to be in the lower
half-plane $a<0$, therefore a possible blowup can happen only for $a<0$ (see
Figure 1, left);

2\. Since
$\dfrac{da}{db}=-\dfrac{a^{2}+b}{(1-b)a}-\epsilon\dfrac{f(1-b)}{1-b}\leqslant-\dfrac{a^{2}+b}{(1-b)a}$,
and $b$ decreases with $t$ as $a<0$, then the Chaplygin theorem implies that
the phase curve of the perturbed equation, $a_{\epsilon}(b)$ lies higher that
the phase curve of the non-perturbed equation,
$a_{0}(b)=-\sqrt{1-2b+C(1-b)^{2}}.$ Therefore for $C<0$ the curve
$a_{\epsilon}(b)$ always comes back to the upper half-plane $a>0$. So, for a
possible blow-up we have to consider only the initial data corresponding to
$C\geqslant 0$ (see Figure 1, right).

3\. Analogously, the Chaplygin theorem implies that if the data are such that
the phase curve $a_{\epsilon_{1}}(b)$ does not go to infinity, then
$a_{\epsilon_{2}}(b)$, $\epsilon_{2}>\epsilon_{1}$ does not go to infinity as
well. Therefore, we can consider for the proof arbitrarily small $\epsilon$.

Since the perturbation by means of parameter $\epsilon$ is regular, in a
neighborhood $U_{\epsilon}(0)$ we can expand the solution in a series

$a_{\epsilon}(b)=a_{0}(b)+\sum\limits_{k=1}^{\infty}\epsilon^{k}\alpha_{k}(b)=a_{0}(b)+\epsilon\alpha_{1}(b)+o(\epsilon),$

converging at any fixed $b$.

Thus, if the first corrector $\alpha_{1}(b)$ is such that
$a_{0}(b)+\epsilon\alpha_{1}(b)>0$ for some $b_{*}$ and arbitrary small
positive $\epsilon$, then we can guarantee that $a_{\epsilon}(b_{*})>0$, in
other words, the trajectory came back to the upper half-plane. The linear
equations for the correctors are the following:

$\displaystyle\dfrac{d\alpha_{1}}{db}$ $\displaystyle=$
$\displaystyle-\alpha_{1}Q-\dfrac{f(1-b)}{1-b},$ (10)
$\displaystyle\dfrac{d\alpha_{k}}{db}$ $\displaystyle=$
$\displaystyle-\alpha_{k}Q+\dfrac{b\phi_{k}(a_{0},\alpha_{1},\dots,\alpha_{k-1})}{(1-b)a_{0}^{k}},\quad
k=2,\dots,$ (11)

where
$Q=\dfrac{a_{0}^{2}-b}{(1-b)a_{0}^{2}}=\dfrac{1-3b+C(1-b)^{2}}{(1-3b+C(1-b)^{2})(1-b)}$,
and $\phi_{k}$ is a homogeneous polynomial of order $k$ from its arguments,
$a_{0}$ is found in (9). It can be readily found from (10) that

$\alpha_{1}(b)=\frac{(1-b)^{2}}{a_{0}(b)}\int\limits_{b}^{b_{0}}\dfrac{a_{0}(\beta)f(1-\beta)}{(1-\beta)^{3}}\,d\beta>0,\quad
b<b_{0}=\rm const.$

As for $C>0$

$a_{0}(b)+\epsilon\alpha_{1}(b)\sim-\sqrt{C}(1-b)+\epsilon\frac{(1-b)^{2}}{a_{0}(b)}\int\limits_{b}^{b_{0}}\dfrac{f(1-\beta)}{(1-\beta)^{2}}\,d\beta,\quad
b\to-\infty,$

then condition (5) guarantees boundedness of the phase trajectory
$a_{\epsilon}(b)$. If the integral (5) converges, then for sufficiently small
$\epsilon$ the prevailing term is $a_{0}(b)$, and the trajectory goes to
infinity. Moreover, for sufficiently small $\epsilon$, satisfying condition

$-\sqrt{C}+\epsilon\int\limits_{\eta_{0}>0}^{+\infty}\dfrac{f(\eta)}{\eta^{2}}\,d\eta<0,$
(12)

the solution blowup for the same initial data as in the non-perturbed case.

For $C=0$ the analysis is analogous, but the result is different. Namely,
$a_{0}(b)\sim-(1-b)^{\frac{3}{2}}$ as $b\to-\infty$ and the respective
condition for boundedness of trajectory is

$\lim\limits_{\eta\to\infty}\eta\int\limits_{\eta_{0}>0}^{\eta}\dfrac{f(\tilde{\eta})}{\tilde{\eta}^{{5/2}}}\,d\tilde{\eta}=\infty.$
(13)

Condition (13) is predictively more mild than (5). For example, for
$f(\eta)=\eta^{\gamma}$, (13) gives $\gamma>\frac{1}{2}$, whereas (5) gives
$\gamma\geqslant 1$. Nevertheless, we have to take into account the worst
situation, i.e. $C>0$.

Figure 2 shows the effect of $\epsilon$ on the solution. It can be seen that
even for sufficiently large $\epsilon$ the solution first closely mimics the
unperturbed case and only after some time sharply changes its behavior (Figure
2, left). In fact, for the threshold value $\gamma=1$ and sufficiently small
$\epsilon$, the difference between the perturbed and unperturbed cases is at
first so small that it cannot be detected numerically. Figure 2, right, shows
rapidly decaying oscillations for a sufficiently long time.

Thus, since the analytical solution to (2), (3) is unique, if the data belong
to the class (4), so does the solution. The theorem is proved. $\Box$

Figure 1:  $f(n)=n^{2}$. Left: the direction field to system (6),
$\epsilon=0.8$. Right: the phase curves starting from the same point for (8)
at $\epsilon=0$, singularity formation (dash) and $\epsilon=0.8$, smooth
solution (solid).

Figure 2: Solution of (6) for $f(n)=n^{2}$. Left: the behavior of $b(t)$ at
$\epsilon=0$, singularity formation (dash) and $\epsilon=1$, smooth solution
(solid). Right: the behavior of $b(t)$ for $\epsilon=1$ near equilibrium
$b=0$.

##  3 Arbitrary initial data

######  Theorem 3.1.

(Main theorem) Let $f(n)\in{\mathcal{A}}({\mathbb{R}}_{+})$ be a nonnegative
function satisfying conditions

$\lim\limits_{\eta\to\infty}\dfrac{\eta f^{\prime}(\eta)}{f(\eta)}=\rm const$
(14)

and (5), $\nu(n)=\epsilon f(n)$, $\epsilon=\rm const>0$. Under the assumption
that the formation of singularity is associated with a gradient catastrophe
(unboundedness of the first derivatives), the problem (2), (3) admits a global
in time classical ($C^{1}$-smooth) solution. Otherwise, one can find the data
such that the derivatives of solution blow up in a finite time. Moreover, the
solution $(V,E)$ itself remains bounded for all nonnegative $f(n)$.

###### Proof.

We denote $q=V_{x}$, $s=E_{x}$, $\xi=V_{xx}$, $\sigma=E_{xx}$ and
differentiate (2) with respect to $x$. Since $n=1-s>0$, then it makes sense to
consider only the half-plane $s<1$. Along every characteristic line $x(t)$,
starting from point $x_{0}\in\mathbb{R}$ we get system

$\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle-E-\epsilon f(1-s)V,$
(15) $\displaystyle\dot{E}$ $\displaystyle=$ $\displaystyle V,$ (16)
$\displaystyle\dot{q}$ $\displaystyle=$
$\displaystyle-q^{2}-s-\epsilon(f(1-s)q-Vf^{\prime}(1-s)\sigma),$ (17)
$\displaystyle\dot{s}$ $\displaystyle=$ $\displaystyle(1-s)q,$ (18)

complemented by initial conditions $V(0)=V(x_{0})$, $E(0)=E(x_{0})$,
$q(0)=V_{x}(x_{0})$, $s(0)=E_{x}(x_{0})$.

First of all we notice that (15), (16) imply

$\displaystyle\frac{d}{dt}(V^{2}+E^{2})=-2f(1-s)V^{2}\leqslant 0,$ (19)

therefore $V$ and $E$, being smooth, remain bounded.

Due to the term $V\,f^{\prime}(1-s)\,\sigma$ system (15) - (18) is not closed.
Indeed, we can write similar equation for $\sigma$, however it contains $\xi$
and $\sigma_{x}$. Thus, continuing differentiation, we cannot obtain a finite
system of derivatives of the solution. It is the principal difficulty
comparing with the case $\epsilon=0$, treated in [5]. For the solutions (4)
this problem does not arise, since $\sigma=0$ for them.

Further, (17), (18) imply

$\displaystyle\dfrac{dq}{ds}=-\dfrac{q^{2}+s}{(1-s)q}-\epsilon\frac{f(1-s)}{1-s}+\epsilon\frac{\sigma
f^{\prime}(1-s)}{(1-s)q}V,$ (20)

which coincides with (8), except of the last term.

As in the proof of Theorem 2.1, we are going to prove that under certain
conditions the projection of the phase curve of system (15) - (18) to the
plane $(s,q)$ is bounded. This would mean that the derivatives of solution of
the problem (2), (3) remain bounded. To this end, we are going to show that
the last term of (20) is subjected to the previous one as $s\to-\infty$ and
then apply the arguments of Theorem 2.1. Variables $(s,q)$ correspond to
$(a,b)$ in the proof of Theorem 2.1.

First of all, we introduce a new independent variable as $s=s(t)$. This is
possible if $\dot{s}\neq 0$, i.e $q\neq 0$. The blow-up implies that $s$ tends
to $\infty$ as $t\to t_{*}<\infty$, for $q<0$.

We then consider the expansion in terms of the regular small parameter
$\epsilon$.

Let us set $q(s)=q_{0}(s)+\epsilon q_{1}(s)+o(\epsilon),$
$\sigma(s)=\sigma_{0}(s)+\epsilon\sigma_{1}(s)+o(\epsilon),$
$\xi(s)=\xi_{0}(s)+\epsilon\xi_{1}(s)+o(\epsilon),$ fixed $s$. Then we find
from (20) as in (9)

$\displaystyle q_{0}(s)=\pm\sqrt{1-2s+C(1-s)^{2}},\quad
C=\frac{q_{0}^{2}(0)+2s(0)-1}{(1-s(0))^{2}}$ (21)

and

$\displaystyle\dfrac{dq_{1}}{ds}$ $\displaystyle=$ $\displaystyle-
q_{1}Q(s)-\dfrac{f(1-s)}{1-s}+\dfrac{\sigma_{0}f^{\prime}(1-s)V(s,q_{0})}{(1-s)q_{0}},$
(22)

where
$Q(s)=\dfrac{q_{0}^{2}-s}{(1-s)q_{0}^{2}}=\dfrac{1-3s+C(1-s)^{2}}{(1-3s+C(1-s)^{2})(1-s)}$.

To find $\sigma_{0}$, we differentiate (2) twice with respect to $x$, write
the equations along the characteristic lines $x(t)$, replace the independent
variable by $s$ and obtain a system of linear equations

$\displaystyle\dfrac{d\sigma_{0}}{ds}$ $\displaystyle=$
$\displaystyle\dfrac{(1-s)\xi_{0}-\sigma_{0}q_{0}}{(1-s)q_{0}},$ (23)
$\displaystyle\dfrac{d\xi_{0}}{ds}$ $\displaystyle=$
$\displaystyle-\dfrac{2q_{0}\xi_{0}+\sigma_{0}}{(1-s)q_{0}}.$ (24)

Further, taking into account (21), from system (23), (24) we get

$\displaystyle\sigma_{0}(s)$ $\displaystyle=$
$\displaystyle(s-1)(C_{1}s+C_{2}q_{0}(s)),$ (25)

with constants $C_{1},C_{2}$, depending on
$s(0),q_{0}(0),\sigma_{0}(0),\xi_{0}(0)$. Note that $\sigma_{0}$, $\xi_{0}$
remain bounded for $s$ bounded.

Due to (25) and condition (14) the ratio
$\dfrac{\sigma_{0}f^{\prime}(1-s)}{(1-s)q_{0}}$ is equivalent to
$\dfrac{f(1-s)}{(1-s)}$ as $s\to-\infty$.

Now we are going to prove that $V(s,q_{0}(s))$ tends to zero as $s\to-\infty$
and therefore the behavior of $q_{1}(s)$, given by (22), is defined only by
term $\dfrac{f(1-s)}{1-s}$.

From (16) – (17) we have

$\displaystyle\frac{dV(s)}{ds}$ $\displaystyle=$
$\displaystyle\frac{-E(s)-\epsilon f(1-s)V(s)}{(1-s)q(s)},$
$\displaystyle\frac{dE(s)}{ds}$ $\displaystyle=$
$\displaystyle\frac{V(s)}{(1-s)q(s)}.$

Since $V(s,q(s))=V(s,q_{0}(s))+O(\epsilon)$,
$E(s,q(s))=E(s,q_{0}(s))+O(\epsilon)$, $\epsilon\to 0$, fixed $s$, then for
the zero terms $V_{0}(s)=V(s,q_{0}(s))$, $E_{0}(s)=E(s,q_{0}(s))$ we obtain

$\displaystyle\frac{dV_{0}}{ds}$ $\displaystyle=$
$\displaystyle\frac{-E_{0}-\epsilon f(1-s)V_{0}}{(1-s)q_{0}},$ (26)
$\displaystyle\frac{dE_{0}}{ds}$ $\displaystyle=$
$\displaystyle\frac{V_{0}}{(1-s)q_{0}}.$ (27)

Further, for convenience, we change the independent variable once again as

$\displaystyle
s_{1}(s)=-\arctan\frac{s}{\sqrt{1-2s+C(1-s)^{2}}}+\arctan\frac{1}{\sqrt{C}}\sim\frac{1}{\sqrt{C}(1-s)},\quad
s\to\infty.$

Then $s_{1}\to 0+$ and $f(1-s)\sim f((\sqrt{C}s_{1})^{-1})$ as $s\to\infty$.
With the new independent variable (26), (27) take the form

$\displaystyle\frac{dV_{0}(s_{1})}{ds_{1}}$ $\displaystyle=$ $\displaystyle
E_{0}(s_{1})+\epsilon f(1-s)V_{0}(s_{1}),$ (28)
$\displaystyle\frac{dE_{0}(s_{1})}{ds_{1}}$ $\displaystyle=$ $\displaystyle-
V_{0}(s_{1}).$ (29)

Let us study the structure of the solution near the point $s_{1}=0$. From (28)
and (29) we obtain the second order Lienard equation

$\displaystyle\frac{d^{2}E_{0}(s_{1})}{ds_{1}^{2}}-\epsilon
f(1-s)\frac{dE_{0}(s_{1})}{ds_{1}}+E_{0}(s_{1})=0,$ (30)

$f(1-s)\sim f((\sqrt{C}s_{1})^{-1})$, $s_{1}\to 0$, and take into account that
(14) implies that there exists $\gamma=\rm const$ such that
$f(\eta)=f_{0}\eta^{\gamma},$ $\eta\to\infty$.

First, we assume that $\gamma\in\mathbb{N}$ and look for the first terms of
expansion into power series as
$E_{0}(s_{1})=E_{\infty}+E_{1}s_{1}^{m}+o(s_{1}^{m})$,
$V_{0}(s_{1})=V_{\infty}+V_{1}s_{1}^{k}+o(s_{1}^{k})$, $s_{1}\to 0$, with
constant $E_{\infty}$, $E_{1}$, $V_{\infty}$, $V_{1}$, and $m,k\in\mathbb{N}$
(recall that $V$ and $E$ are bounded).

If we substitute the expansion of $E$ into (30), we get

$\displaystyle
E_{\infty}-E_{1}m(m-1)s_{1}^{m-2}+K_{1}s_{1}^{m-\gamma-1}+o(s_{1}^{m_{*}})=0,$
(31)

where $m_{*}=\min\\{m-2,m-\gamma-1\\}$ and $K_{1}$ is a new constant depending
on $\epsilon,E_{1},m,C,f_{0}$. If $\gamma>1$, then the lower term is
$s_{1}^{m-\gamma-1}$, it should be equalized with $E_{\infty}s_{1}^{0}$.
Therefore $E_{\infty}=K_{1}\neq 0$ and $m=\gamma+1$, $E_{1}\neq 0$ can be
expressed in terms of $E_{\infty}$. Analogously, if $\gamma=1$, then the lower
term is $s_{1}^{m-2}$. Therefore $m=2$ and $E_{\infty}=2E_{1}+K$. It rests to
substitute $E_{0}=E_{\infty}+E_{1}s_{1}^{m}+o(s_{1}^{m})$ into (29) and get
$V_{0}=E_{1}ms_{1}^{m-1}+o(s_{1}^{m-1})$, therefore $V_{\infty}=0$,
$k=\gamma$.

To prove the previous result for any $\gamma>0$, it suffices to prove it for
all rational $\gamma$. If $\gamma=\dfrac{p}{r}\in\mathbb{Q}$,
$p,r\in\mathbb{N}$, then to obtain in the right-hand side of (30) a rational
function, we set $s_{1}=s_{2}^{r}$, rewrite system (30) it terms of this new
variable and look for a solution
$E_{0}(s_{2})=E_{\infty}+E_{1}s_{2}^{m}+o(s_{2}^{m})$,
$V_{0}(s_{2})=V_{\infty}+V_{1}s_{2}^{k}+o(s_{2}^{k})$, $s_{2}\to 0$, with
constant $E_{\infty}$, $E_{1}$, $V_{\infty}$, $V_{1}$, and $m,k\in\mathbb{N}$.
The result of substitution is

$\displaystyle
E_{\infty}-\frac{E_{1}m(m-1)}{r^{2}}s_{2}^{m-2r}+K_{2}s_{2}^{m-p-r}+o(s_{1}^{m_{*}})=0,$
(32)

where $m_{*}=\min\\{m-2r,m-p-1\\}$ and $K_{2}$ is a new constant depending on
$\epsilon,E_{1},m,r,C,f_{0}$. We reason as above and obtain $m=p+r$. Therefore
$E_{0}(s_{1})=E_{\infty}+E_{1}s_{1}^{\gamma+1}+O(s_{1}^{\gamma})$,
$V_{0}(s_{1})=V_{1}s_{1}^{\gamma}+o(s_{1}^{\gamma})$, $s_{1}\to 0$. Thus, in
all cases $V_{\infty}=0$ and $V(s)\to 0$ as $s\to-\infty$.

As for the higher-order terms, $q_{i}$, $i=2,\dots$, they obey a system of
linear equations, similar to (11), therefore for a fixes $s$, the property to
come back in the upper half-plane $q>0$ is defined only by $q_{1}(s)$ for
sufficiently small $\epsilon$. However at any $\epsilon>0$ for $s,q\to-\infty$
the last term in (20) is subjected to the previous one, therefore, equation
(20) is equivalent at the point of singularity formation to (8), and the phase
trajectory for any initial data turns out in the upper half-plane $q>0$, where
$q,s$ cannot go to infinity. Theorem 3.1 is proved. $\Box$

######  Remark 1.

We notice that for the non-perturbed case $\epsilon=0$ the derivatives of
solution can go to infinity at $V_{\infty}=\lim\limits_{s=-\infty}V(s),$ this
value is defined by initial data and it can be any constant. For the case
$\epsilon>0$, the blow-up necessarily happens for $V_{\infty}=0$.

######  Remark 2.

From the very beginning, one can suspect that the last term in (20) can be
neglected near the point of singularity formation. Indeed,
$\sigma=E_{xx}=-n_{x}$, the density $n$ is positive and reaches its maximum at
the point of the singularity.

######  Remark 3.

The conditions of the theorem do not forbid the formation of a weak
discontinuity of the solution along a specific characteristic. Moreover, in
place of this weak discontinuity, in principle, after some time, the
unboundedness of the first derivatives can form. As far as we know, such
interesting and subtle phenomena have not yet been investigated.

##  4 Discussion

1\. System (2) has the form $U_{t}+A(U,U_{x})U_{x}=F(U)$, $U=(V,E)$. The
matrix $A$ is a Jordan block, it has multiple eigenvalue $V$, but only one
eigenvector. The system does not belong to symmetric hyperbolic one, and
nonlinear resonance can occure in the solution [7]. The simplest system of
this form is the so-called pressureless gas dynamics. It is commonly known
that there the component of density develops the delta-singularity. In our
case it happens at the points where $E_{x}$ tends to $-\infty$. At the same
time, the pressureless gas dynamics system is much simpler than the one
considered here, since for smooth solutions the equation for the velocity
component is separated, and the singularity formation process can be easily
studied.

2\. There exist different approaches to the well-posedness of weak solutions
to the pressureless Euler-Poisson equations [7], [8], [9].

3\. For the Euler analog of system (1),

$\dfrac{\partial V}{\partial t}+(V\cdot\nabla)V=-\nu(n)V,\quad\dfrac{\partial
n}{\partial t}+{\rm div}\,(nV)=0,$

the result about the possibility of finding such a friction factor $f(n)$ to
suppress singularity formation is not valid. Thus, the elliptic coupling is
crucial for the effectiveness of the density-dependent damping.

## Acknowledgements

The author thanks Evgenyi Chizhonkov and Alexander Kurganov for stimulation
discussion and numerical computations helping to clarify the structure of
solutions.

## References

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# Euler Products at the centre and applications to Chebyshev’s Bias

Arshay Sheth  Mathematics Institute, Zeeman Building, University of Warwick,
Coventry, CV4 7AL, UK<EMAIL_ADDRESS>https://arshaysheth.github.io

###### Abstract.

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ with associated $L$-function
$L(s,\pi)$. We study the behaviour of the partial Euler product of $L(s,\pi)$
at the center of the critical strip. Under the assumption of the Generalized
Riemann Hypothesis for $L(s,\pi)$ and assuming the Ramanujan–Petersson
conjecture when necessary, we establish an asymptotic, off a set of finite
logarithmic measure, for the partial Euler product at the central point that
confirms a conjecture of Kurokawa. As an application, we obtain results
towards Chebyshev’s bias in the recently proposed framework of Aoki-Koyama
[AK23] .

The author is supported by funding from the European Research Council under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 101001051 — Shimura varieties and the Birch–Swinnerton-Dyer
conjecture).

##  1\. Introduction

In this paper, we study the behaviour of partial Euler products of automorphic
$L$-functions at the central point of their critical strip. The motivation for
investigating the behaviour of Euler products at the central point dates back
to the original version of the Birch and Swinnerton-Dyer conjecture.

######  Conjecture 1.1 (Birch and Swinnerton-Dyer [BSD65]).

Let $E/\mathbb{Q}$ be an elliptic curve with rank $r$ and for each prime $p$,
let $N_{p}=\\#E_{\mathrm{ns}}(\mathbb{F}_{p})$, where
$E_{\mathrm{ns}}(\mathbb{F}_{p})$ denotes the set of non-singular
$\mathbb{F}_{p}$-rational points on a minimal Weierstrass model for $E$ at
$p$. Then we have that

$\prod_{p\leq x}\frac{N_{p}}{p}\sim C(\log x)^{r}$

as $x\to\infty$ for some constant $C$ depending on $E$.

This conjecture can be viewed as an assertion about the asymptotics of the
partial Euler product at the central point $s=1$ of the $L$-function $L(E,s)$
attached to $E$. If we let $N_{E}$ denote the conductor of $E$ and define
$a_{p}=p+1-N_{p}$ for $p\nmid N_{E}$ and let $a_{p}=p-N_{p}$ for $p|N_{E}$,
$L(E,s)$ is defined for $\mathrm{Re}(s)>\frac{3}{2}$ by

$L(E,s)\coloneqq\prod_{p\mid N_{E}}\frac{1}{1-a_{p}p^{-s}}\prod_{p\nmid
N_{E}}\frac{1}{1-a_{p}p^{-s}+p^{1-2s}}.$

By the work of Wiles [Wil95] and Breuil–Conrad–Diamond–Taylor [BCDT01],
$L(E,s)$ admits an analytic continuation to the the entire complex plane and
has a functional equation relating values at $s$ and $2-s$. Defining

$P_{E}(x)=\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}\frac{1}{1-a_{p}p^{-1}}\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid N_{E}\end{subarray}}\frac{1}{1-a_{p}p^{-1}+p^{-1}}$

to be the partial Euler product at $s=1$, Conjecture 1.1 can be reformulated
to assert that

$P_{E}(x)\sim\frac{1}{C(\log x)^{r}}$

as $x\to\infty$. Conjecture 1.1 has since motivated the study of partial Euler
products of $L$-functions in their critical strip. For instance, if $\chi$ is
a non-trivial Dirichlet character with associated Dirichlet $L$-function
$L(s,\chi)$, Conrad has shown (see [Con05, Theorem 3.3]) that the equality

$\lim\limits_{x\to\infty}\prod\limits_{p\leq
x}(1-\chi(p)p^{-s})^{-1}=L(s,\chi)$ (1.1)

for all $s$ with $\textrm{Re}(s)>\frac{1}{2}$ is equivalent to the Generalized
Riemann Hypothesis for $L(s,\chi)$. More generally, in the case of entire
$L$-functions, Conrad showed that the convergence of the Euler product in the
right-half of the critical strip is equivalent to the Generalized Riemann
Hypothesis for the $L$-function.

It is natural to investigate the convergence of the Euler product on the
critical line as well. It is believed that, except for zeros on the critical
line, Euler products of entire $L$-functions should also converge everywhere
on the critical line, and the limit of the Euler product at a point on the
line should equal the value of the $L$-function at the point i.e. the analog
of Equation (1.1) should also hold everywhere on the critical line except on
the set containing the zeros of the $L$-function. However, at the central
point, there is often an unexpected factor of $\sqrt{2}$ that is known to
appear in the Euler product asymptotics.

For instance, Goldfeld [Gol82] showed that if Conjecture 1.1 is true, then
$C=\frac{r!}{L^{(r)}(E,1)}\cdot\sqrt{2}e^{r\gamma}$, where $\gamma$ is Euler’s
constant; thus if $P_{E}(x)$, the partial Euler product at the centre,
converges to a non-zero value as $x\to\infty$, then its value is
$L(E,1)/\sqrt{2}$ (as opposed to simply $L(E,1)$). A conceptual explanation of
the unexpected appearance of $\sqrt{2}$ was subsquently given by Conrad in
terms of second-moment $L$-functions. If an $L$-function $L(s)$, which we
henceforth assume is normalized so its centre is at $s=\frac{1}{2}$, is given
by an Euler product

$L(s)=\prod_{p}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1},$

its second moment $L$-function is given by

$L_{2}(s)=\prod_{p}\prod_{j=1}^{n}(1-\alpha_{j,p}^{2}p^{-s})^{-1}$

and in practice is the ratio of the corresponding symmetric square
$L$-function and the exterior square $L$-function. Let
$R=\mathop{\mathrm{ord}}\limits_{s=1}L_{2}(s)$; Conrad showed that if the
Euler product at the center converges, then its value equals
$L(\frac{1}{2})/\sqrt{2}^{R}$.

######  Example 1.2.

If $\chi$ is a Dirichlet character, $L_{2}(\chi,s)=L(\chi^{2},s)$. Hence, if
$\chi$ is a quadratic character, then $R=-1$; thus for a quadratic character,
if $\lim\limits_{x\to\infty}\prod\limits_{p\leq x}(1-\chi(p)p^{-1/2})^{-1}$
exists, then

$\lim_{x\to\infty}\prod_{p\leq x}(1-\chi(p)p^{-1/2})^{-1}=\sqrt{2}\cdot
L\left(\frac{1}{2},\chi\right).$

Based on the above phenomena Kurokawa et al. (see for instance [KKK22] or
[KKK14]) formulated a general conjecture about the convergence of partial
Euler products at the centre of the critical strip. This conjecture has been
called the “Deep Riemann Hypothesis”, since it not only implies the
Generalized Riemann Hypothesis but, as we explain below, seems in a precise
sense to lie deeper than the Generalized Riemann Hypothesis.

We now briefly explain this conjecture in the setting of general automorphic
$L$-functions attached to an irreducible cuspidal automorphic representation
$\pi$ of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. We choose to work in this
very general setting since in the applications in Section 4 of the paper, we
consider a wide range of $L$-functions (attached to Dirichlet characters,
elliptic curves and modular forms), and it is useful for us to have a general
framework that encompasses these different $L$-functions. Indeed, according to
the Langlands conjectures, the most general $L$-functions can all be expressed
as products of $L$-functions attached to cuspidal automorphic representations
of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$, and all the $L$-functions
needed in our applications are known to be examples of automorphic
$L$-functions. Any automorphic $L$-function (see Section 2) can be written in
the form

$L(s,\pi)=\prod_{p}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1},$

where the $\alpha_{j,p}$’s are the Satake parameters for the local
representation $\pi_{p}$. We let
$\nu(\pi)=m(\mathrm{sym}^{2}\pi)-m(\wedge^{2}\pi)\in\mathbb{Z}$, where
$m(\rho)$ denotes the multiplicity of the trivial representation 1 in $\rho$.
If we let $L_{2}(s,\pi)$ denote the second-moment $L$-function associated to
$L(s,\pi)$, we have that $\nu(\pi)=-R(\pi)$, where as above
$R(\pi):=\mathop{\mathrm{ord}}_{s=1}L_{2}(s,\pi)$. Throughout this paper, we
also assume that $L(s,\pi)$ is entire.

######  Conjecture 1.3 (Kaneko-Koyama-Kurokawa [KKK22]).

Keep the assumptions and notation as above. Let
$m=\mathop{\mathrm{ord}}_{s=1/2}L(s,\pi)$. Then the limit

$\lim_{x\to\infty}\left((\log x)^{m}\prod_{p\leq
x}\prod_{j=1}^{n}\left(1-\alpha_{j,p}p^{-\frac{1}{2}}\right)^{-1}\right)$
(1.2)

satisfies the following conditions:

(A):

The limit (1.2) exists and is nonzero.

(B):

The limit (1.2) satisfies

$\lim_{x\to\infty}\left((\log x)^{m}\prod_{p\leq
x}\prod_{j=1}^{n}\left(1-\alpha_{j,p}p^{-\frac{1}{2}}\right)^{-1}\right)=\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\frac{1}{2},\pi\right).$

By truncating the Euler product at large $x$ for a wide range of
$L$-functions, there is now numerical evidence for Conjecture 1.3 (see for
instance [Con05, page 281] or [KKK14, Section 3]). As another piece of
evidence, we also mention that the function field analog of the conjecture has
been proven by Kaneko–Koyama–Kurokawa (see [KKK22, Theorem 5.2]). To explain
the relation between Conjecture 1.3 and the Generalized Riemann Hypothesis
(for the $L$-function $L(s,\pi)$), let
$a_{\pi}(p^{k})=\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k}$ and let

$\psi(x,\pi)=\sum_{p^{k}\leq x}\log p\cdot a_{\pi}(p^{k}).$

Then the Generalized Riemann Hypothesis is equivalent to the fact that

$\psi(x,\pi)=O(\sqrt{x}(\log x)^{2})$ (1.3)

while Conjecture 1.3 is equivalent to the estimate

$\psi(x,\pi)=o(\sqrt{x}\log x).$ (1.4)

It is in this sense that Conjecture 1.3 seems deeper than the Generalized
Riemann Hypothesis. The estimate in Equation (1.4) is indeed plausible; an
analysis of Montgomery [Mon70] about the vertical distribution of zeros of
$L$-functions on the critical line suggests that the true order of magnitude
of $\psi(x,\pi)$ is at most $O(\sqrt{x}(\log\log\log x)^{2})$ which would
imply (1.4).

However, error terms may not be the best way to determine to the precise
relation between the Generalized Riemann Hypothesis and Conjecture 1.3; for
instance, the Generalized Riemann Hypothesis is also equivalent to the
slightly weaker error term $\psi(x,\pi)=O(x^{\frac{1}{2}+\epsilon})$ for any
$\epsilon>0$. Since Conjecture 1.3 concerns the convergence of the Euler
product at the centre of the critical strip, it is natural to ask whether the
Generalized Riemann Hypothesis can be related to the Euler product at the
centre as well. Our first result answers this question in the affirmative, by
showing that in fact the Generalized Riemann Hypothesis implies Conjecture 1.3
outside a set of finite logarithmic measure. We first recall the following
definition.

######  Definition 1.4.

Let $S\subseteq\mathbb{R}_{\geq 2}$ be a measurable subset of the real
numbers. The logarithmic measure of $S$ is defined to be

$\mu^{\times}(S)=\int_{S}\frac{dt}{t}.$

We remark that since we are working in this level of generality, we also need
to assume the Ramanujan–Petersson conjecture for $\pi$ (we refer to Section 2
for the description of this conjecture and for the cases it has been proven).

######  Theorem A (Theorem 3.8).

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that $L(s,\pi)$ is entire and
let $m=\mathop{\mathrm{ord}}\limits_{s=\frac{1}{2}}L(s,\pi)$. Assume the
Ramanujan–Petersson Conjecture and the Riemann Hypothesis for $L(s,\pi)$. Then
there exists a subset $S\subseteq\mathbb{R}_{\geq 2}$ of finite logarithmic
measure such that for all $x\not\in S$,

$(\log x)^{m}\cdot\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}})^{-1}\sim\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\frac{1}{2},\pi\right).$

The method of proof first consists of developing a suitable version of an
explicit formula for $L(s,\pi)$ (Proposition 3.1) that allows us to establish
the asymptotic behaviour of partial Euler products of $L(s,\pi)$ in the right-
half of the critical strip (Theorem 3.2) . One of the terms in this asymptotic
formula involves contributions coming from the zeros of $L(s,\pi)$; this term
is the most delicate to handle, and the standard bound in Equation (1.3)
coming from the Generalized Riemann Hypothesis will not suffice. Rather, we
need the refined estimate $\psi(x,\pi)=O(\sqrt{x}(\log\log x)^{2})$ which
holds, conditional on the Generalized Riemann Hypothesis, outside a set of
finite logarithmic measure (Theorem 3.4). The method described here builds
upon previous work of the author [She23], where these techniques were used to
study the relations between the original and modern formulations of the Birch
and Swinnerton–Dyer conjecture. In this paper, we further explore applications
of these ideas to questions concerning Chebyshev’s bias.

###  1.1. Applications to Chebyshev’s bias

Chebyshev’s bias originally referred to the phenomenon that, even though the
primes are equidistributed in the multiplicative residue classes mod 4, there
seem to more primes congruent to 3 mod 4 than 1 mod 4. Let $\pi(x;q,a)$ denote
the number of primes up to $x$ which are congruent to $a$ modulo $q$ and let
$S=\\{x\in\mathbb{R}_{\geq 2}:\pi(x;4,3)-\pi(x;4,1)>0\\}$. Building on
Chebyshev’s observations, Knapowski–Turán [KT62] conjectured that the
proportion of postive real numbers which lie in the set $S$ would equal 1 as
$x\to\infty$. However, this conjecture was later disproven by Kaczorowski
[Kac95] conditionally on the Generalized Riemann Hypothesis, by showing that
the limit does not exist. Rubinstein and Sarnak [RS94] instead considered the
logarithmic density $\delta(S):=\frac{1}{\log
X}\cdot\lim\limits_{X\to\infty}\int_{t\in S\cap[2,X]}\frac{dt}{t}$ of $S$;
assuming the Generalized Riemann Hypothesis and the assumption that the non-
negative imaginary parts of zeros of Dirichlet $L$-functions are linearly
independent over $\mathbb{Q}$, they showed that this limit exists and
$\delta(S)=0.9959\ldots$, hence giving a satisfactory explanation of this
phenomenon. Similar biases have since been observed in various other
situations as well. For instance, if $E/\mathbb{Q}$ is an elliptic curve and
$a_{p}$ denotes the trace of the Frobenius at the prime $p$, then even though
$a_{p}$ is positive and negative equally often by the Sato–Tate conjecture,
Mazur [Maz08] noted $D_{E}(x)=\\#\\{p\leq x:a_{p}>0\\}-\\#\\{p\leq
x:a_{p}<0\\}$ has a bias towards being negative if the rank of $E$ is large.
An explanation of this fact was subsquently given by Sarnak [Sar07] in the
spirit of [RS94]. A conceptual framework for dealing with problems related to
Chebyshev’s bias, generalizing the Rubinstein–Sarnak approach to a wide range
of $L$-functions, was recently given by Devin [Dev20].

In [AK23], Aoki-Koyama present an alternative approach to describe Chebyshev’s
bias that is closely related to the behaviour of Euler products at the center
of the critical strip.

######  Definition 1.5 (Aoki-Koyama [AK23]).

Let $(c_{p})_{p}\subseteq\mathbb{R}$ be a sequence over primes $p$ such that

$\lim_{x\to\infty}\frac{\\#\\{p\mid c_{p}>0,p\leq x\\}}{\\#\\{p\mid
c_{p}<0,p\leq x\\}}=1.$

We say that $(c_{p})_{p}$ has a Chebyshev bias towards being positive if there
exists a positive constant $C$ such that $\displaystyle{\sum_{p\leq
x}\frac{c_{p}}{\sqrt{p}}\sim C\log\log x}$. On the other hand, we say that
$c_{p}$ is unbiased if $\displaystyle{\sum_{p\leq
x}\frac{c_{p}}{\sqrt{p}}=O(1)}.$

For instance, if we let $\chi_{4}$ to be the non-trivial Dirichlet character
modulo $4$, $c_{p}=\chi_{4}(p)$, then the sum in Definition 1.5 becomes
$\sum_{p\leq
x}\frac{\chi(p)}{\sqrt{p}}=\pi_{\frac{1}{2}}(x;4,1)-\pi_{\frac{1}{2}}(x;4,3),$
where
$\displaystyle{\pi_{s}(x;q,a)=\sum\limits_{\begin{subarray}{c}p<x\colon\text{prime}\\\
p\equiv a{\@displayfalse\,(\mathrm{mod}{\,q})}\end{subarray}}\frac{1}{p^{s}}}$
for $s\geq 0$ is the weighted prime counting function.

Using Theorem A, we obtain the following asymptotic for the types of sums
appearing in Definition 1.5.

######  Theorem B (Theorem 4.2).

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that $L(s,\pi)$ is entire and
let $m=\mathop{\mathrm{ord}}_{s=1/2}L(s,\pi)$. Assume the Ramanujan–Petersson
Conjecture and the Riemann Hypothesis for $L(s,\pi)$. Then there exists a
constant $c_{\pi}$ such that

$\sum_{p\leq
x}\frac{\alpha_{1,p}+\cdots+\alpha_{n,p}}{\sqrt{p}}=\left(\frac{R(\pi)}{2}-m\right)\log\log
x+c_{\pi}+o(1)$

for all $x$ outside a set of finite logarithmic measure, where
$R(\pi)=\mathop{\mathrm{ord}}\limits_{s=1}L_{2}(s,\pi)$.

As applications of this theorem, we obtain results towards Chebyshev’s bias,
in the sense of Definition 1.5, for a wide class of equidistributed sequences
in number theory.

######  Corollary A (Corollary 4.3).

Let $\chi_{4}$ denote the non-trivial Dirichlet character modulo 4. Assume the
Riemann Hypothesis for $L(\chi_{4},s)$. Then there exists a constant $c$ such
that

$\sum_{p\leq x}\frac{\chi_{4}(p)}{\sqrt{p}}=-\frac{1}{2}\log\log x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure. In particular, in the
sense of Definition 1.5, there is a Chebyshev bias towards primes which are 3
mod 4.

######  Corollary B (Corollary 4.4).

Let $\tau(n)$ denote Ramanujan’s tau function. Assume the Riemann Hypothesis
for $L(s,\Delta)$. Then there exists a constant $c$ such that

$\sum_{p\leq x}\frac{\tau(p)}{p^{6}}=\frac{1}{2}\log\log x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure. In particular, in the
sense of Definition 1.5, the sequence $\tau(p)p^{-\frac{11}{2}}$ has a
Chebyshev bias towards being positive.

######  Corollary C (Corollary 4.5).

Let $E/\mathbb{Q}$ be an elliptic curve with rank $\mathrm{rk}(E)$ and
Frobenius trace $a_{p}$ for each prime $p$. Assume the Birch and Swinnerton-
Dyer conjecture and the Riemann Hypothesis for $L(E,s)$. Then there exists a
constant $c$ depending on $E$ such that

$\sum_{p\leq x}\frac{a_{p}}{p}=\left(\frac{1}{2}-\mathrm{rk}(E)\right)\log\log
x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure. In particular,
$a_{p}$ has a bias towards being positive if $\mathrm{rk}(E)=0$ and a bias
towards being negative if $\mathrm{rk}(E)>0$.

Corollary C is a concrete instance of Birch and Swinnerton-Dyer’s initial
observations, which motivated them to formulate their celebrated conjecture,
that elliptic curves which have more rational points also tend to have more
than the expected number of points modulo primes $p$.

Corollaries A, B and C have been inspired by similar statements in [AK23],
[KK22] and [KK23] respectively. The key difference is that the statements in
op.cit. assume Conjecture 1.3, but our statements only assume the Generalized
Riemann Hypothesis, with the caveat that our asymptotics hold for all $x$
outside an exceptional set. Indeed, as far as we are aware, this is the first
instance in the literature where an explanation of the above mentioned
phenomenona can be given only assuming the Generalized Riemann Hypothesis; as
mentioned above, previous work on the subject, such as the Rubinstein–Sarnak
approach, also assume deep conjectures about the linear independence of zeros
of the relevant $L$-functions. Finally, we refer the reader to [AK23] for more
examples of this flavour such as Chebyshev’s bias in the splitting of prime
ideals in Galois extension of number fields; we also note that there is
related work by Okumura [Oku23], studying Chebyshev’s bias for Fermat curves
of primes degree. Using Theorem B, analogs for all these results can be proven
outside a finite set of logarithmic measure assuming only the Generalized
Riemann Hypothesis.

### Acknowledgements

I would like to thank Shin-ya Koyama for asking the questions that led to this
paper and for helpful suggestions. I would also like to thank Adam Harper and
Nuno Arala Santos for helpful discussions.

##  2\. Preliminary background

In this section, we review the properties of automorphic $L$-functions that we
will need in the rest of the paper. Let $\pi=\bigotimes^{\prime}\pi_{v}$ be an
irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. Outside a finite set of places,
for each finite place $p$, $\pi_{p}$ is unramified and we can associate to
$\pi_{p}$ a semisimple conjugacy class $\\{A_{\pi}(p)\\}$ in
$\mathrm{GL}_{n}(\mathbb{C})$. Such a conjugacy class is parameterized by its
eigenvalues $\alpha_{1,p},\ldots,\alpha_{n,p}$. The local Euler factors
$L_{p}(s,\pi_{p})$ are given by

$L_{p}(s,\pi_{p})=\det(1-A_{\pi}(p)p^{-s})^{-1}=\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}.$

At the ramified finite primes, the local factors are best described by the
Langlands parameters of $\pi_{p}$ (see for instance the Appendix in [SR96]).
They are of the form $L_{p}(s,\pi_{p})=P_{p}(p^{-s})^{-1}$, where $P_{p}(x)$
is a polynomial of degree at most $n$, and $P_{p}(0)$ = 1. We will in this
case too write the local factors in the form above, with the convention that
we now allow some of the $\alpha$’s to be zero. The global $L$-function
attached to $\pi$ is given by

$\displaystyle L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})$
$\displaystyle=\prod_{p}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1},$

By the works of Godement–Jacquet [GJ72] and Jacquet–Shalika [JS81], $L(s,\pi)$
defines a holomorphic function for $s\in\mathbb{C}$ with $\mathrm{Re}(s)>1$
and admits a meromorphic continuation to the entire complex plane. For
$\mathrm{Re}(s)>1$ we have that

$-\frac{L^{\prime}(s,\pi)}{L(s,\pi)}=\sum_{n=1}^{\infty}\frac{\Lambda(n)a_{\pi}(n)}{n^{s}},$
(2.1)

where $\Lambda(n)$ is the von-Mangoldt function and

$a_{\pi}(p^{k})=\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k}.$

Sarnak and Rudnick [SR96] have shown that if $\pi_{p}$ is unramified, then
$|\alpha_{j,p}|<p^{\frac{1}{2}-\frac{1}{n^{2}+1}}$. In this paper, we will
often assume the following stronger bound.

######  Conjecture 2.1 (The Ramanujan-Petersson Conjecture).

For any $p$ such that $\pi_{p}$ is uramified, $|\alpha_{j,p}|=1$ for all
$j\in\\{1,\ldots,n\\}$.

The Ramanujan-Petersson Conjecture is known in certain cases (for instance
when $n=1$ and by the works [Del74] and [DS74], for all modular $L$-functions
of degree two). The completed $L$-function of $L(s,\pi)$ is given by
$\Lambda(s,\pi)=Q(\pi)^{s/2}L_{\infty}(s,\pi)L(s,\pi),$ where $Q(\pi)$ is the
conductor of the representation and the archimedean factor is given by
$L_{\infty}(s,\pi)=\prod_{j=1}^{n}\Gamma_{\mathbb{R}}(s+\mu_{\pi}(j)),$ where
the $\mu_{\pi}(j)$’s are certain constants and
$\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2)$. The completed $L$-function
satisfies a functional equation

$\Lambda(s,\pi)=\epsilon(\pi)\Lambda(1-s,\pi^{\vee}),$

where $\epsilon(\pi)\in\mathbb{C}$ is a complex number of absolute one and
$\pi^{\vee}$ is the contragradient representation of $\pi$. The set of trivial
zeros of $L(s,\pi)$ is $\\{-2k-\mu_{\pi}(j):k\in\mathbb{N}\text{ and
}j\in\\{1,\ldots,n\\}\\}$. The Riemann Hypothesis for $L(s,\pi)$ is the
statement that all the non-trivial zeros of $L(s,\pi)$ are on the line
$\textrm{Re}(s)=\frac{1}{2}$.

##  3\. Proof of Theorem A

Throughought this section, we let $\pi$ be an irreducible cuspidal automorphic
representation of $\textrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that the
associated $L$-function $L(s,\pi)$ is entire. We begin by establishing a
suitable version of an explicit formula for $L(s,\pi)$.

######  Proposition 3.1.

Let $s\in\mathbb{C}\setminus{\\{\frac{1}{2}\\}}$ be a complex number such that
$L(s,\pi)\neq 0$. We have that

$\sum_{n\leq
x}\frac{\Lambda(n)a_{\pi}(n)}{n^{s}}=-m\cdot\frac{x^{\frac{1}{2}}-s}{\frac{1}{2}-s}-\frac{L^{\prime}(s,\pi)}{L(s,\pi)}-\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}+\sum_{j=1}^{n}\sum_{k=0}^{\infty}\frac{x^{-2k-\mu_{\pi}(j)-s}}{2k+\mu_{\pi}(j)+s},$

where $m=\mathop{\mathrm{ord}}_{s=\frac{1}{2}}L(s,\pi)$, the sum over $\rho$
is taken over all non-trivial zeros of $L(s,\pi)$ (excluding
$\rho=\frac{1}{2}$) counting multiplicity and is interpreted as
$\lim_{T\to\infty}\sum_{|\gamma|\leq T}$, and where the last term of the sum
on the left hand side is weighted by half if $x$ is an integer.

###### Proof.

By using Perron’s formula and Equation (2.1), we have that

$\sum_{n\leq x}\frac{\Lambda(n)a_{\pi}(n)}{n^{s}}=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}\left(\sum_{n=1}^{\infty}\frac{\Lambda(n)a_{\pi}(n)}{n^{s+z}}\right)\frac{x^{z}}{z}dz=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}-\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\frac{x^{z}}{z}dz$

for $c\in\mathbb{R}$ sufficiently large. By shifting the contour to the left
and applying Cauchy’s residue theorem, the above integral equals the sum of
residues of the integrand in the region $\mathrm{Re}(z)\leq c$.

  * •

When
$\displaystyle{s+z=\frac{1}{2},\mathop{\mathrm{res}}_{z=\frac{1}{2}-s}\left(\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\right)=m}$,
so
$\displaystyle{\mathop{\mathrm{res}}_{z=\frac{1}{2}-s}\left(-\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\frac{x^{z}}{z}\right)=-m\cdot\frac{x^{\frac{1}{2}}-s}{\frac{1}{2}-s}}$.

  * •

When $z=0$,
$\displaystyle{\mathop{\mathrm{res}}_{z=0}\left(-\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\frac{x^{z}}{z}\right)=-\frac{L^{\prime}(s,\pi)}{L(s,\pi)}}$.

  * •

If $\rho\neq\frac{1}{2}$ is a non-trivial zero of $L(s,\pi)$,
$\displaystyle{\mathop{\mathrm{res}}_{z=\rho-s}\left(-\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\frac{x^{z}}{z}\right)=-r_{\rho}\cdot\frac{x^{\rho-s}}{\rho-s}}$,
where $r_{\rho}$ is the order of the zero $\rho$. Thus, we obtain a total
contribution of
$\displaystyle{-\sum\limits_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}}$
counting multiplicity for all the non-trivial zeros of $L(s,\pi)$.

  * •

Finally, the trivial zeros of $L(s,\pi)$ are at
$\\{-2k-\mu_{\pi}(j):k\in\mathbb{N}\text{ and }j\in\\{1,\ldots,n\\}\\}$. Thus,
since
$\displaystyle{\mathop{\mathrm{res}}_{z=-2k-\mu_{\pi}(j)-s}\left(-\frac{L^{\prime}(s+z,\pi)}{L(s+z,\pi)}\frac{x^{z}}{z}\right)=\frac{x^{-2k-\mu_{\pi}(j)-s}}{2k+\mu_{\pi}(j)+s}}$
for all $k\geq 0$ and $j\in\\{1,2,\ldots,n\\}$, we get a total contribution of
$\displaystyle{\sum_{j=1}^{n}\sum_{k=0}^{\infty}\frac{x^{-2k-\mu_{j}(\pi)-s}}{2k+\mu_{j}(\pi)+s}}$.

∎

We now note that

$\displaystyle\frac{d}{ds}\left(\log\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}\right)=-\sum_{j=1}^{n}\sum_{p\leq
x}\frac{\log p\cdot\alpha_{j,p}}{p^{s}-\alpha_{j,p}}$

and for each $j\in\\{1,\ldots,n\\}$, we write

$\sum_{p\leq x}\frac{\log
p\cdot\alpha_{j,p}}{p^{s}-\alpha_{j,p}}=\sum_{\begin{subarray}{c}p\leq
x\end{subarray}}\frac{\log
p\cdot\alpha_{j,p}}{p^{s}}+\sum_{\begin{subarray}{c}p\leq
x\end{subarray}}\frac{\log p\cdot\alpha_{j,p}^{2}}{p^{2s}}+\sum_{k\geq
3}\sum_{\begin{subarray}{c}p\leq x\end{subarray}}\frac{\log
p\cdot\alpha_{j,p}^{k}}{p^{ks}}.$

It thus follows that

$\displaystyle\frac{d}{ds}$ $\displaystyle\left(\log\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}\right)=-\sum_{j=1}^{n}\sum_{p\leq
x}\frac{\log p\cdot\alpha_{j,p}}{p^{s}-\alpha_{j,p}}=-\sum_{n\leq
x}\frac{\Lambda(n)a_{\pi}(n)}{n^{s}}$
$\displaystyle-\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}\frac{\log
p\cdot(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{p^{2s}}-\sum_{k\geq
3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\end{subarray}}}\frac{\log
p\cdot(\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k})}{p^{ks}}.$

######  Theorem 3.2.

Assume the Ramanujan–Petersson Conjecture and the Riemann Hypothesis for
$L(s,\pi)$. Then for a complex number $s\in\mathbb{C}$ with
$\frac{1}{2}<\mathrm{Re}(s)<1$, we have that

$\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}=L(s,\pi)\exp\left(-m\cdot\mathrm{Li}(x^{\frac{1}{2}-s})-R_{s}(x)+U_{s}(x)+O\left(\frac{\log
x}{x^{1/6}}\right)\right),$

where we have set

  * •

$m=\mathop{\mathrm{ord}}\limits_{s=\frac{1}{2}}L(s,\pi)$,

  * •

$\mathrm{Li}(x)$ is the principal value of
$\displaystyle{\int\limits_{0}^{x}\frac{dt}{\log t}}$

  * •

$\displaystyle{R_{s}(x)=\frac{1}{\log
x}\sum\limits_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}+\frac{1}{\log
x}\sum\limits_{\rho\neq\frac{1}{2}}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz}$

  * •

$\displaystyle{U_{s}(x)=\sum\limits_{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}\frac{(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{2p^{2s}}}$.

Here, the sums in the term $R_{s}(x)$ are taken over all non-trivial zeros
$\rho=\frac{1}{2}+i\gamma$ of $L(s,\pi)$ (excluding $\rho=\frac{1}{2}$)
counted with multiplicity and are interpreted as
$\lim_{T\to\infty}\sum_{|\gamma|\leq T}$, and the integral is taken along the
horizontal line starting at $s$.

###### Proof.

For all $s\in\mathbb{C}$ with $\frac{1}{2}<\mathrm{Re}(s)<1$, we have using
Proposition 3.1 and the previous equation that

$\displaystyle\frac{d}{ds}$ $\displaystyle\left(\log\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}\right)=m\cdot\frac{x^{\frac{1}{2}}-s}{\frac{1}{2}-s}+\frac{L^{\prime}(s,\pi)}{L(s,\pi)}+\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}-\sum_{j=1}^{n}\sum_{k=0}^{\infty}\frac{x^{-2k-\mu_{\pi}(j)-s}}{2k+\mu_{\pi}(j)+s}$
$\displaystyle-\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}\frac{\log
p\cdot(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{p^{2s}}-\sum_{k\geq
3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\end{subarray}}}\frac{\log
p\cdot(\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k})\cdot}{p^{ks}}$ .

We now fix $s_{0}\in\mathbb{C}$ with $\frac{1}{2}<\mathrm{Re}(s)<1$.
Integrating this equation along the horizontal straight line from $s_{0}$ to
$\infty$, it follows that

$\displaystyle\log\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s_{0}})^{-1}$
$\displaystyle=-m\int_{s_{0}}^{\infty}\frac{x^{\frac{1}{2}}-s}{\frac{1}{2}-s}-\int_{s_{0}}^{\infty}\frac{L^{\prime}(s,\pi)}{L(s,\pi)}-\sum_{\rho\neq\frac{1}{2}}\int_{s_{0}}^{\infty}\frac{x^{\rho-s}}{\rho-s}$
$\displaystyle+\int_{s_{0}}^{\infty}\sum_{j=1}^{n}\sum_{k=0}^{\infty}\frac{x^{-2k-\mu_{\pi}(j)-s}}{2k+\mu_{\pi}(j)+s}+\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{p^{2s}}+$
$\displaystyle+\sum_{k\geq 3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq
x\end{subarray}}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k})\cdot}{p^{ks}}$

These integrals can now be analysed in an analogous way as in [She23, Theorem
2.3] to obtain the desired formula for the partial Euler product in the
critical strip. ∎

######  Remark 3.3.

The implied constant in the big O term above (and in all big O terms appearing
in the paper) depends only the automorphic representation $\pi$ and is
independent of $s$.

Let $\psi(x,\pi)=\sum_{n\leq x}\Lambda(n)a_{n}(\pi)$. It is known that, under
the Riemann Hypothesis for $L(s,\pi)$,

$\psi(x,\pi)=O(\sqrt{x}(\log x)^{2}).$

Using a method introduced by Gallagher in [Gal80], it is possible to improve
the error term outside a set of finite logarithmic measure.

######  Theorem 3.4.

Let $\pi$ be an irreducible cuspidal representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that $L(s,\pi)$ is entire.
Assume the Riemann Hypothesis for $L(s,\pi)$. Then there exists a set $S$ of
finite logarithmic measure such that for all $x\not\in S$,

$\psi(x,\pi)\ll\sqrt{x}(\log\log x)^{2}.$

###### Proof.

The analog of this result for the prime counting function $\psi(x)$ was first
proven by Gallagher [Gal80] and was later generalized in the setting of the
theorem by Qu ([Qu07, Theorem 1.1]); similar arguments have also been given in
[Koy16, Theorem 2] and [She23, Theorem 3.4]. ∎

As mentioned in the introduction, we let
$R(\pi)=\mathop{\mathrm{ord}}_{s=1}L_{2}(s,\pi)$ where $L_{2}(s,\pi)$ is the
second-moment $L$-function associated to $L(s,\pi)$ and is defined by

$L_{2}(s,\pi)=\prod_{p}\prod_{j=1}^{n}(1-\alpha_{j,p}^{2}p^{-s})^{-1}.$

As explained in [Dev20, Example 1], there exists an open subset
$U\supseteq\\{s\in\mathbb{C}:\mathrm{Re}(s)\geq 1\\}$ such $L_{2}(s,\pi)$ can
be continued to a meromorphic function on $U$; thus, $R(\pi)$ is well-defined.
We recall that we set
$\nu(\pi)=m(\mathrm{sym}^{2}\pi)-m(\wedge^{2}\pi)\in\mathbb{Z}$, where
$m(\rho)$ denotes the multiplicity of the trivial representation 1 in $\rho$
and that $\nu(\pi)=-R(\pi)$.

######  Lemma 3.5.

There is a constant $M$ such that

$\sum_{p\leq
x}\frac{\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2}}{p}=-R(\pi)\log\log
x+M+o(1).$

###### Proof.

This follows from the work of Conrad, see [Con05, page 275]. ∎

######  Remark 3.6.

Lemma 3.5 can be regarded as a generalization of Merten’s estimate

$\sum_{p\leq x}\frac{1}{p}=\log\log x+M+o(1).$

Indeed, if all the $\alpha_{i,p}=1$ so that $L_{2}(s,\pi)$ is the Riemann zeta
function, we recover Merten’s estimate.

Lemma 3.5 is essentially the main reason why $\sqrt{2}$ shows up in the Euler
product asymptotics.

######  Corollary 3.7.

Let $U_{s}(x)$ be as in Theorem 3.2. We have that

$\lim_{x\to\infty}U_{\frac{1}{2}}(x)=-R(\pi)\cdot\log\sqrt{2}=\nu(\pi)\cdot\log\sqrt{2}$

###### Proof.

We have that

$\displaystyle U_{\frac{1}{2}}(x)$ $\displaystyle=\sum_{p\leq
x}\frac{\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2}}{2p}-\sum_{p\leq\sqrt{x}}\frac{\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2}}{2p}$
$\displaystyle=\left(-\frac{R(\pi)}{2}\log\log
x+M+o(1)\right)-\left(-\frac{R(\pi)}{2}\log\log\sqrt{x}+M+o(1)\right)$
$\displaystyle=-R(\pi)\cdot\log\sqrt{2}+o(1).$

∎

We can now prove our first main theorem.

######  Theorem 3.8 (Theorem A).

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that $L(s,\pi)$ is entire and
let $m=\mathop{\mathrm{ord}}_{s=1/2}L(s,\pi)$. Assume the Ramanujan–Petersson
conjecture and the Riemann Hypothesis for $L(s,\pi)$. Then there exists a
subset $S\subseteq\mathbb{R}_{\geq 2}$ of finite logarithmic measure such that
for all $x\not\in S$,

$(\log x)^{m}\cdot\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}})^{-1}\sim\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\pi,\frac{1}{2}\right).$

###### Proof.

We follow the method of [She23, Theorem 4.2]. When
$s=\frac{1}{2}+\frac{1}{x}$, the left-hand side of Theorem 3.2 equals

$\displaystyle\prod_{p\leq x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}$
$\displaystyle=\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}-\frac{1}{x}})^{-1}$
$\displaystyle=\prod_{p\leq
x}\prod_{j=1}^{n}\left(1-\alpha_{j,p}p^{-\frac{1}{2}}\left(1+O\left(\frac{\log
p}{x}\right)\right)\right)^{-1}$ $\displaystyle=\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}})^{-1}\cdot\prod_{p}\prod_{j=1}^{n}\left(1+O\left(\frac{\alpha_{j,p}\cdot\log
p}{\sqrt{p}x}\right)\right),$

where we used the fact that

$(1+a+O(f(x)))^{-1}=(1+a)^{-1}(1+O(f(x)))\textrm{ if }f(x)=o(1)\textrm{ and
}a\textrm{ is sufficiently small. }$

Now

$\prod_{p\leq x}\prod_{j=1}^{n}\left(1+O\left(\frac{\alpha_{j,p}\cdot\log
p}{\sqrt{p}x}\right)\right)=1+O\left(\sum_{p\leq
x}\frac{|\alpha_{j,p}|\cdot\log p}{\sqrt{p}x}\right)=1+o(1),$

where we used the Ramanujan–Petersson Conjecture and the estimate $\sum_{p\leq
x}\frac{1}{\sqrt{p}}\ll\frac{\sqrt{x}}{\log x}$. Thus, when
$s=\frac{1}{2}+\frac{1}{x}$,

$\prod_{p\leq x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-s})^{-1}\sim\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}})^{-1}.$ (3.1)

We now estimate the right-hand side of Theorem 3.8. Write

$L(s,\pi)=a_{m}\left(s-\frac{1}{2}\right)^{m}+a_{m+1}\left(s-\frac{1}{2}\right)^{m+1}+\cdots$

so that when $s=\frac{1}{2}+\frac{1}{x}$, $L(s,\pi)\sim
a_{m}\cdot\frac{1}{x^{m}}$ as $x\to\infty$.

To estimate the contribution coming from the term
$\mathrm{Li}(x^{\frac{1}{2}-s})$, we use the classical fact (see for instance
[Fin03, pp.425] or [Har40, Equation (2.2.5)]) that

$\mathrm{Li}(x)=\gamma+\log|\log x|+\sum_{n=1}^{\infty}\frac{(\log
x)^{n}}{n!\cdot n}\textrm{ for all }x\in\mathbb{R}_{>0}\setminus\\{1\\}.$

Applying this when $s=\frac{1}{2}+\frac{1}{x}$ yields that

$\mathrm{Li}(x^{\frac{1}{2}-s})=\gamma+\log\left(\frac{\log
x}{x}\right)+o(1).$

To estimate the contribution coming from the term $U_{s}(x)$ we note that
$U_{\frac{1}{2}+\frac{1}{x}}(x)\to U_{\frac{1}{2}}(x)$ as $x\to\infty$ since,
by a similar argument as above, we have that

$\displaystyle\sum_{{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}}\frac{(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{2p^{2(\frac{1}{2}+\frac{1}{x})}}$
$\displaystyle=\sum_{{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}}\frac{(\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2})}{2p}\cdot\left(1+O\left(\frac{\log
p}{x}\right)\right)$ $\displaystyle=\sum_{{\begin{subarray}{c}\sqrt{x}<p\leq
x\end{subarray}}}\frac{(\alpha_{1,p}^{2}+\cdots\alpha_{n,p}^{2})}{2p}+O\left(\frac{1}{x}\sum_{\sqrt{x}<p\leq
x}\frac{\log p}{p}\right)=U_{\frac{1}{2}}(x)+o(1).$

Using Corollary 3.7, we conclude that when $s=\frac{1}{2}+\frac{1}{x}$,
$U_{s}(x)\to\nu(\pi)\cdot\log\sqrt{2}$ as $x\to\infty$.

To estimate the contribution coming from the term

$R_{s}(x)=\frac{1}{\log
x}\sum\limits_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}+\frac{1}{\log
x}\sum\limits_{\rho\neq\frac{1}{2}}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz$

when $s=\frac{1}{2}+\frac{1}{x}$, we begin by noting that

$\sum_{\rho\neq\frac{1}{2}}\frac{1}{\rho-s}-\sum_{\rho\neq\frac{1}{2}}\frac{1}{\rho}=\sum_{\rho\neq
1}\frac{s}{(\rho-s)\rho}$

and that

$\sum_{\rho\neq\frac{1}{2}}\frac{s}{(\rho-s)\rho}\ll\sum_{\rho}\frac{1}{|\rho|^{2}}<\infty,$

where the fact that above sum converges follows, for instance, from the
discussion in [Dev20, Remark 1]. Thus,

$\sum_{\rho\neq\frac{1}{2}}\frac{1}{\rho-s}=\sum_{\rho}\frac{1}{\rho}+O(1)$

and, since we are assuming the Riemann Hypothesis for $L(s,\pi)$,

$\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}=\frac{1}{x^{s}}\sum_{\rho}\frac{x^{\rho}}{\rho}+O(x^{\rho-s})\ll\frac{1}{\sqrt{x}}\sum_{\rho}\frac{x^{\rho}}{\rho}+O(1).$

Using the Riemann von-Mangoldt explicit formula for $L(s,\pi)$ (see [Qu07,
Theorem 3.1])

$\psi(x,\pi)=-\sum_{\rho}\frac{x^{\rho}}{\rho}+O(x^{\frac{1}{2}-\frac{1}{n^{2}+1}}\log
x),$

it follows that

$\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}\ll\frac{1}{\sqrt{x}}\psi(x,\pi)+O(1).$

By Theorem 3.4, we conclude that there exists a set $S$ of finite logarithmic
measure such that for all $x\not\in S$,

$\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}\ll\frac{1}{\sqrt{x}}O(\sqrt{x}(\log\log
x)^{2})+O(1)$

and so for all $x\not\in S$,

$\frac{1}{\log x}\sum_{\rho\neq\frac{1}{2}}\frac{x^{\rho-s}}{\rho-s}=o(1).$

To estimate the second quantity in the definition of $R_{s}(x)$, we again use
the fact that $\sum\limits_{\rho}\frac{1}{|\rho|^{2}}$ converges to conclude
that

$\displaystyle\frac{1}{\log
x}\sum_{\rho\neq\frac{1}{2}}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz$
$\displaystyle=O\left(\frac{1}{\log
x}\cdot\int_{s}^{\infty}x^{\frac{1}{2}-\mathrm{Re}(z)}d|z|\right)$
$\displaystyle=O\left(\frac{x^{\frac{1}{2}-\textrm{Re}(s)}}{\log^{2}x}\right)=o(1).$

Thus, in summary, when $s=\frac{1}{2}+\frac{1}{x}$, we conclude that for all
$x\not\in S$ we have that

$\displaystyle
L(s,\pi)\exp\left(-m\cdot\mathrm{Li}(x^{\frac{1}{2}-s})-R_{s}(x)+U_{s}(x)+O\left(\frac{\log
x}{x^{1/6}}\right)\right)$
$\displaystyle=L(s,\pi)\exp\left(-m\gamma-m\log\left(\frac{\log
x}{x}\right)+\nu(\pi)\cdot\log\sqrt{2}+o(1)\right)$
$\displaystyle\sim\frac{a_{m}}{x^{m}}\cdot\exp\left(-m\gamma-m\log\left(\frac{\log
x}{x}\right)+\nu(\pi)\cdot\log\sqrt{2}\right)=\frac{1}{(\log
x)^{m}}\cdot\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\frac{1}{2},\pi\right)$

Combining this with Equation (3.1) proves the theorem. ∎

##  4\. Applications to Chebyshev’s Bias

In this section, we use Theorem 3.8 to obtain results towards Chebyshev’s bias
in the framework of [AK23]. We recall the following definition.

######  Definition 4.1.

Let $(c_{p})_{p}\subseteq\mathbb{R}$ be a sequence over primes $p$ such that

$\lim_{x\to\infty}\frac{\\#\\{p\mid c_{p}>0,p\leq x\\}}{\\#\\{p\mid
c_{p}<0,p\leq x\\}}=1.$

We say that $(c_{p})_{p}$ has a Chebyshev bias towards being positive if there
exists a positive constant C

$\sum_{p\leq x}\frac{c_{p}}{\sqrt{p}}\sim C\log\log x.$

On the other hand, we say that $c_{p}$ is unbiased if

$\sum_{p\leq x}\frac{c_{p}}{\sqrt{p}}=O(1).$

As mentioned in the introduction, the sum appearing in this definition is
closely related to partial Euler products of $L$-functions at the centre of
the critical strip. Indeed, using Theorem 3.8, we obtain the following
asymptotic.

######  Theorem 4.2 (Chebyshev’s bias for Satake parameters).

Let $\pi$ be an irreducible cuspidal automorphic representation of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ such that $L(s,\pi)$ is entire and
let $m=\mathop{\mathrm{ord}}_{s=1/2}L\left(s,\pi\right)$. Assume the Riemann
Hypothesis and the Ramanujan-Petersson Conjecture for $L(s,\pi)$. Then there
exists a constant $c_{\pi}$ such that

$\sum_{p\leq
x}\frac{\alpha_{1,p}+\cdots+\alpha_{n,p}}{\sqrt{p}}=\left(\frac{R(\pi)}{2}-m\right)\log\log
x+c_{\pi}+o(1)$

for all $x$ outside a set of finite logarithmic measure, where
$R(\pi)=\mathop{\mathrm{ord}}\limits_{s=1}L_{2}(s,\pi)$.

###### Proof.

By Theorem 3.8, we have that

$(\log x)^{m}\cdot\prod_{p\leq
x}\prod_{j=1}^{n}(1-\alpha_{j,p}p^{-\frac{1}{2}})^{-1}=\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\frac{1}{2},\pi\right)+o(1)$

for all $x$ outside a set $S$ of finite logarithmic measure. Taking logarithms
yields that

$m\log\log x+\sum_{p\leq
x}\frac{\alpha_{1,p}+\cdots+\alpha_{n,p}}{\sqrt{p}}+\sum_{p\leq
x}\frac{\alpha_{1,p}^{2}+\cdots+\alpha_{n,p}^{2}}{2p}+\sum_{p\leq
x}\sum_{k\geq
3}\frac{\alpha_{1,p}^{k}+\cdots+\alpha_{n,p}^{k}}{kp^{k/2}}=C+o(1)$

for all $x\not\in S$, where
$C=\log\left(\frac{\sqrt{2}^{\nu(\pi)}}{e^{m\gamma}m!}\cdot
L^{(m)}\left(\frac{1}{2},\pi\right)\right)$. Using Lemma 3.5 and the fact that
the last term on the left-hand side above converges, we get as desired that

$\sum_{p\leq
x}\frac{\alpha_{1,p}+\cdots+\alpha_{n,p}}{\sqrt{p}}=\left(\frac{R(\pi)}{2}-m\right)\log\log
x+c_{\pi}+o(1)$

for some constant $c_{\pi}$. ∎

We now proceed to record a number of special cases of the above theorem.

###  4.1. The mod $4$ prime number race

######  Corollary 4.3.

Let $\chi_{4}$ denote the non-trivial Dirichlet character modulo 4. Assume the
Riemann Hypothesis for $L(\chi_{4},s)$. Then there exists a constant $c$ such
that

$\sum_{p\leq x}\frac{\chi_{4}(p)}{\sqrt{p}}=-\frac{1}{2}\log\log x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure. In particular,

$\pi_{\frac{1}{2}}(x;4,3)-\pi_{\frac{1}{2}}(x;4,1)=\frac{1}{2}\log\log
x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure.

###### Proof.

It is well known that $L(\chi_{4},\frac{1}{2})\neq 0$ and by Example 1.2,
$R(\chi_{4})=-1$. The result thus follows from Theorem 4.2. ∎

###  4.2. Chebyshev’s bias for Ramanujan’s $\tau$ function

We consider the $\Delta$ function defined by

$\Delta(q)=q\prod_{n=1}^{\infty}(1-q^{n})^{24}=\sum_{n=1}^{\infty}\tau(n)q^{n}$

where $q=e^{2\pi iz}$ and
$z\in\mathbb{H}=\\{z\in\mathbb{C}:\mathrm{Im}(z)>0\\}$. The $L$-function
attached to $\Delta$ is defined to be

$L(s,\Delta)=\sum_{n=1}^{\infty}\frac{\tau(n)}{n^{s}}=\prod_{p}(1-\tau(p)p^{-s}+p^{11-2s})^{-1}\hskip
14.22636pt\text{for }\mathrm{Re}(s)>\frac{13}{2}$

and has an analytic continuation to the entire complex plane.

######  Corollary 4.4.

Assume the Riemann Hypothesis for $L(s,\Delta)$. Then there exists a constant
$c$ such that

$\sum_{p\leq x}\frac{\tau(p)}{p^{6}}=\frac{1}{2}\log\log x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure.

###### Proof.

By the work of Deligne [Del74] we have that $|\tau(p)|\leq 2p^{11/2}$, so we
can write $\tau(p)=2p^{11/2}\cos(\theta_{p})$ for a unique
$\theta_{p}\in[0,\pi]$. Let $\pi_{\Delta}$ denote the automorphic
representation corresponding to $\pi$; then we have that
$\displaystyle{L(s,\pi_{\Delta})=L\left(s+\frac{11}{2},\Delta\right)}.$ An
elementary calculation shows that we have

$L\left(s+\frac{11}{2},\Delta\right)=\prod_{p}(1-e^{i\theta_{p}}p^{-s})^{-1}(1-e^{-i\theta_{p}}p^{-s})^{-1}.$

It is known that $L\left(s,\pi_{\Delta}\right)\neq 0$ and $R(\pi_{\Delta})=1$
(see [KK22]). Thus, the asymptotic follows from Theorem 4.2 since

$\sum_{p\leq x}\frac{e^{i\theta_{p}}+e^{-i\theta_{p}}}{\sqrt{p}}=\sum_{p\leq
x}\frac{2\cos(\theta_{p})}{\sqrt{p}}=\sum_{p\leq x}\frac{\tau(p)}{p^{6}}.$

∎

###  4.3. Chebyshev’s bias for Frobenius traces of elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N_{E}$ with Frobenius
trace $a_{p}$ for each prime $p$ and rank $\textrm{rk}(E)$.

######  Corollary 4.5.

Assume the Riemann Hypothesis for $L(E,s)$ and the Birch and Swinnerton-Dyer
conjecture. Then there exists a constant $c$ depending on $E$ such that

$\sum_{p\leq x}\frac{a_{p}}{p}=\left(\frac{1}{2}-\mathrm{rk}(E)\right)\log\log
x+c+o(1)$

for all $x$ outside a set of finite logarithmic measure.

###### Proof.

By the Hasse-Weil bound, we can write $a_{p}=2\sqrt{p}\cos(\theta_{p})$ for a
unique $\theta_{p}\in[0,\pi]$. As before, we have that the normalized
$L$-function is given by

$L\left(E,s+\frac{1}{2}\right)=\prod_{p|N_{E}}(1-a_{p}p^{-s-\frac{1}{2}})^{-1}\cdot\prod_{p\nmid
N_{E}}(1-e^{i\theta_{p}}p^{-s})^{-1}(1-e^{-i\theta_{p}}p^{-s})^{-1}.$

By [Con05, Example 4.7], the second moment $L$-function associated to
$L\left(E,s+\frac{1}{2}\right)$ has a simple zero at $s=1$. Thus, noting that
the contribution from the factors at the bad primes can be absorbed into the
constant, the asymptotic follows from Theorem 4.2 since

$\sum_{{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{e^{i\theta_{p}}+e^{-i\theta_{p}}}{\sqrt{p}}=\sum_{{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}}\frac{2\cos(\theta_{p})}{\sqrt{p}}=\sum_{{\begin{subarray}{c}p\leq
x\\\ p\nmid N_{E}\end{subarray}}}\frac{a_{p}}{p}.$

∎

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