# Euler product Asymptotics for $L$-functions of elliptic curves

Arshay Sheth  Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
UK<EMAIL_ADDRESS>
###### Abstract.

We establish the asymptotic behaviour of partial Euler products of
$L$-functions of elliptic curves in the right-half of their critical strip. As
applications of our result, we study the relations between the Riemann
Hypothesis for $L(E,s)$ and the original as well as modern formulations of the
Birch and Swinnerton-Dyer conjecture. If $E$ is an elliptic curve with
$r:=\mathop{\mathrm{ord}}\limits_{s=1}L(E,s)$, and $N_{p}$ denotes the number
of points of $E$ modulo a prime $p$, we prove that the Riemann Hypothesis for
$L(E,s)$ implies that

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

for all $x$ outside a set of finite logarithmic measure. As a corollary, we
obtain a new proof of Goldfeld’s theorem that the original version of the
Birch and Swinnerton-Dyer conjecture implies the modern formulation of the
conjecture i.e. we show that the asymptotic $\prod\limits_{p\leq
x}\frac{N_{p}}{p}\sim C(\log x)^{\mathrm{rank}(E(\mathbb{Q}))}$ as
$x\to\infty$ implies that
$\mathop{\mathrm{ord}}\limits_{s=1}L(E,s)=\mathrm{rank}(E(\mathbb{Q}))$.

##  1\. Introduction

Even though Euler products of $L$-functions are generally valid only to the
right of the critical strip, there is a strong sense in which they should also
continue to persist inside the critical strip. One of the earliest results
based on this paradigm is the following theorem of Ramanujan.

######  Theorem 1.1 (Ramanujan).

Assume the Riemann Hypothesis for $\zeta(s)$. Let $s\in\mathbb{C}$ be a fixed
complex number with $\frac{1}{2}<\mathrm{Re}(s)<1$. Then we have that

$\prod_{p\leq
x}(1-p^{-s})^{-1}=-\zeta(s)\exp\left(\mathrm{Li}(\vartheta(x)^{1-s})+\frac{2sx^{\frac{1}{2}-s}}{(2s-1)\log
x}+\frac{S_{s}(x)}{\log
x}+O\left(\frac{x^{\frac{1}{2}-s}}{\log(x)^{2}}\right)\right),$

where $\mathrm{Li}(x)$ is the principal value of
$\int\limits_{0}^{x}\frac{dt}{\log t}$, $\vartheta(x)=\sum\limits_{p\leq
x}\log p$ is Chebyshev’s function and
$S_{s}(x)=-s\sum\limits_{\rho}\frac{x^{\rho-s}}{\rho(\rho-s)}$, the sum taken
over all non-trivial zeros of $\zeta(s)$.

As Ramanujan’s formula shows, there is a very close connection (with a
relatively small error term) between $\zeta(s)$ and the (partial) Euler
product of $\zeta(s)$ even inside the critical strip. Ramanujan discovered
this formula in the process of obtaining the maximal order of the divisor
function during his study of highly composite numbers. Ramanujan’s initial
investigations were published in [Ram15], but his complete work on the
subject, including the statement and proof of Theorem 1.1, remained in his
“lost notebook”, and was only published in 1997 in [Ram97]. Ramanujan’s
formula was recently generalized to Dirichlet $L$-functions by Kaneko [Kan22].
In fact, if $\chi$ is a non-trivial Dirichlet character, Conrad has shown (see
[Con05, Theorem 3.3]) that the equality

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

for all $s$ with $\textrm{Re}(s)>\frac{1}{2}$ is equivalent to the Generalized
Riemann Hypothesis for $L(s,\chi)$. Conrad also studied the behaviour of
partial Euler products of $L$-functions on the critical line; for instance, he
showed that if $\chi$ is a quadratic character, and
$\lim\limits_{x\to\infty}\prod\limits_{p\leq x}(1-\chi(p)p^{-1/2})^{-1}$
exists, then

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

Conrad explained the unexpected appearance of the factor of $\sqrt{2}$ in
terms of the associated second moment $L$-function of $L(s,\chi)$ (see [Con05,
Theorem 1.2]).

The main theorem of this paper is to establish the asymptotic behaviour of
partial Euler products of $L$-functions of elliptic curves in the right-half
of the critical strip. To state the theorem, we first introduce the following
notation. Let $E/\mathbb{Q}$ be an elliptic curve with conductor $N_{E}$. For
each prime number $p$, we let $N_{p}$ denote the number of
$\mathbb{F}_{p}$-rational points on a minimal Weierstrass model for $E$ at
$p$; we then define $a_{p}=p+1-N_{p}$ if $p\nmid N$ and $a_{p}=p-N_{p}$ if
$p|N$. The $L$-function of $E$ is defined as

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

We have that $L(E,s)$ defines a holomorphic function for all $s\in\mathbb{C}$
with $\textrm{Re}(s)>3/2$; by the work of Wiles [Wil95] and
Breuil–Conrad–Diamond–Taylor [BCDT01], $L(E,s)$ admits an analytic
continuation to the entire complex plane and satisfies a functional equation
which relates $L(E,s)$ to $L(E,2-s)$. In particular, the critical strip of
$L(E,s)$ is the region $\frac{1}{2}<\mathrm{Re}(s)<\frac{3}{2}$ and the
critical line of $L(E,s)$ is the line $\mathrm{Re}(s)=1$. More precisely, if
we define $\Lambda(E,s)=N_{E}^{s/2}(2\pi)^{-s}\Gamma(s)L(E,s)$ the functional
equation asserts that

$\Lambda(E,s)=w_{E}\Lambda(E,2-s)$

for some $w_{E}\in\\{\pm 1\\}$. Since $\Gamma(s)$ has poles at
$s=0,-1,-2,\ldots$ it follows from the functional equation that $L(E,s)$ has
trivial zeros at $s=0,-1,-2,\ldots$ . The Riemann Hypothesis for $L(E,s)$
asserts that all the non-trivial zeros of $L(E,s)$ lie on the line
$\textrm{Re}(s)=1$.

Our main theorem can now be stated as follows.

######  Theorem 1.2.

Assume the Riemann hypothesis holds for $L(E,s)$. Then for a complex number
$s\in\mathbb{C}$ with $1<\mathrm{Re}(s)<\frac{3}{2}$, we have that

$\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\cdot\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=L(E,s)\exp\left(-r\cdot\mathrm{Li}(x^{1-s})-R_{s}(x)+O\left(\frac{x^{2-2\mathrm{Re}(s)}-x^{1-\mathrm{Re}(s)}}{(2-2\mathrm{Re}(s))\log
x}\right)\right)\\\ $

where $r=\mathop{\mathrm{ord}}_{s=1}L(E,s)$, $\mathrm{Li}(x)$ is the principal
value of $\int\limits_{0}^{x}\frac{dt}{\log t}$, and $R_{s}(x)=\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\frac{x^{\rho-s}}{\rho-s}+\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz$.
Here, the sums are taken over all non-trivial zeros $\rho=1+i\gamma$ of
$L(E,s)$ (excluding $\rho=1$) counted with multiplicity and are interpreted as
$\lim\limits_{T\to\infty}\sum\limits_{|\gamma|\leq T}$, and the integral is
taken along horizontal line starting at $s$.

######  Remark 1.

This formula can be regarded as analogue of Theorem 1.1 for $L$-functions of
elliptic curves. However, an important difference is that the implicit
constant in the big $O$ notation above (and in all the big $O$ terms appearing
throughout the paper), depends only on our fixed elliptic curve $E$ and is
independent of the variable $s$. In particular, $s$ is allowed to depend on
$x$ and this fact will be important for the applications in Section 4 of the
paper.

The key idea of the proof is to establish a suitable version of an explicit
formula for $L(E,s)$.

We use our main theorem to investigate the relations between the Riemann
Hypothesis for $L(E,s)$ and the original as well as modern formulations of the
Birch and Swinnerton-Dyer conjecture. The original version of the Birch and
Swinnerton-Dyer conjecture (see [BSD65, (A) on page 79]) takes the following
form.

######  Conjecture 1.3 (Birch and Swinnerton-Dyer).

We have that

$\prod_{p\leq x}\frac{N_{p}}{p}\sim C(\log x)^{\mathrm{rank}(E(\mathbb{Q}))}$

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

In [Gol82], Goldfeld proves that the asymptotic $\prod_{p\leq
x}\frac{N_{p}}{p}\sim C(\log x)^{\mathrm{rank}(E(\mathbb{Q}))}$ implies the
Riemann hypothesis for $L(E,s)$. Using Theorem 1.2, we prove a result in the
direction of the converse. We first introduce 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

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

######  Theorem 1.5.

Assume the Riemann Hypothesis for $L(E,s)$. Then there exists a subset
$S\subseteq\mathbb{R}_{\geq 2}$ of finite logarithmic measure such that for
all $x\not\in S$,

$\prod_{p\leq x}\frac{N_{p}}{p}\asymp(\log x)^{r},$

where $r=\mathop{\mathrm{ord}}_{s=1}L(E,s)$. In other words, there exist
positive constants $C_{1}$ and $C_{2}$ such that for all $x\not\in S$,

$C_{1}(\log x)^{r}\leq\prod_{p\leq x}\frac{N_{p}}{p}\leq C_{2}(\log x)^{r}.$

######  Remark 2.

Kuo and Murty [KM05], and independently Conrad [Con05], have shown that if we
let $\alpha_{p}$ and $\beta_{p}$ denote the eigenvalues of the Frobenius
morphism on the Tate-module $T_{\ell}(E)$ (cf. Section 2), and
$\psi_{E}(x)=\sum_{\begin{subarray}{c}p^{k}\leq x\\\ p\nmid
N_{E}\end{subarray}}(\alpha_{p}^{k}+\beta_{p}^{k})\log p$, then Conjecture 1.3
is equivalent to the fact that $\psi_{E}(x)=o(x\log x)$. On the other hand the
Riemann Hypothesis for $L(E,s)$ is equivalent to $\psi_{E}(x)=O(x(\log
x)^{2})$. Thus, Conjecture 1.3 seems deeper than the Riemann Hypothesis for
$L(E,s)$. Theorem 1.5 is consistent with this observation, since we can only
prove the desired order of magnitude for the product $\prod\limits_{p\leq
x}\frac{N_{p}}{p}$ assuming the Riemann Hypothesis for $L(E,s)$, as opposed to
the asymptotic formula in Conjecture 1.3.

To prove Theorem 1.5, we first use a method of Gallagher [Gal80] to obtain,
conditional on the Riemann Hypothesis for $L(E,s)$, the slightly better
estimate $\psi_{E}(x)=O(x(\log\log x)^{2})$ outside a set of finite
logarithmic measure. Theorem 1.5 then follows by letting $s$ tend to one in
Theorem 1.2 and estimating the left and right hand sides of the theorem. As a
corollary to Theorem 1.5, we obtain a new proof of a theorem originally due to
Goldfeld [Gol82] that the original form of the Birch and Swinnerton-Dyer
conjecture implies the modern formulation of the conjecture.

######  Corollary 1.6.

Let $E/\mathbb{Q}$ be an elliptic curve. Suppose that

$\prod_{p\leq x}\frac{N_{p}}{p}\sim C(\log x)^{\mathrm{rank}(E(\mathbb{Q}))}$

as $x\to\infty$. Then
$\mathop{\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}(E(\mathbb{Q})).$

We now briefly compare our methods with those of [Gol82]. Firstly, while
Goldfeld only analyses the asymptotics of the partial Euler product of
$L(E,s)$ at $s=1$ to obtain Corollary 1.6, we obtain the result as a specific
case of our main theorem which analyses, more generally, the asymptotic
behaviour of partial Euler products of $L(E,s)$ in the right-half of the
critical strip. Secondly, our techniques for estimating partial Euler products
are also different from those in [Gol82]: Goldfeld’s analysis involves working
with the second moment $L$-function of $L(E,s)$; this function is defined to
be

$\prod_{p|N_{E}}(1-a_{p}^{2}p^{-s})^{-1}\prod_{p\nmid
N_{E}}(1-\alpha_{p}^{2}p^{-s})^{-1}(1-\beta_{p}^{2}p^{-s})^{-1}$

and is the ratio of the symmetric square $L$-function and the exterior square
$L$-function associated to $L(E,s)$. On the other hand, we prove Corollary 1.6
by working with $L(E,s)$ itself. The explicit formula that we use in this
paper also takes a much simpler shape than the one used in [Gol82]; in
particular, we do not need to use an auxiliary test-function in our version of
the explicit formula. While our approach is simpler, we can make no prediction
about the constant $C$ appearing in Conjecture 1.3. Goldfeld showed that if
Conjecture 1.3 is true, then
$C=\frac{r!}{L^{(r)}(E,1)}\cdot\sqrt{2}e^{r\gamma}$, where $\gamma$ is Euler’s
constant.
We remark that Conrad [Con05] has also given another proof of Corollary 1.6 by
studying how a very general class of Euler products should behave on the
critical line; our approach is also different from Conrad’s approach in two
respects. Firstly, like Goldfeld, Conrad also uses second moment $L$-functions
in his analysis and secondly, Conrad uses Tauberian theorems in lieu of
explicit formulas.

### Notation used in the paper

We write $f=O(g)$ or $f\ll g$ if there exists a positive constant $c$ such
that $|f(z)|\leq c|g(z)|$ for all $z$ in a specified range. As mentioned
above, the constant $c$ is allowed to depend on the elliptic curve $E$. We
write $f\asymp g$ if $f\ll g$ and $g\ll f$.

### Acknowledgements

I would like to thank Adam Harper for very helpful discussions in connection
with this paper and for feedback on a previous draft of this manuscript. I
would also like to thank Nuno Arala Santos for helpful discussions regarding
the material in Section 3 of the paper. This research was supported by funding
from the European Research Council (Grant agreement No. 101001051 — Shimura
varieties and the Birch–Swinnerton-Dyer conjecture).

##  2\. Proof of the main theorem

Let $E/\mathbb{Q}$ be an elliptic curve. Since the Euler factor at a prime $p$
is quadratic for each $p\nmid N_{E}$, it will be convenient for us to
linearise this Euler factor: for each $p\nmid N_{E}$, we let $\alpha_{p}$ and
$\beta_{p}$ denote the roots of the polynomial $x^{2}-a_{p}x+p$ and so each
Euler factor can be written as

$(1-a_{p}p^{-s}+p^{1-2s})=(1-\alpha_{p}p^{-s})\cdot(1-\beta_{p}p^{-s}).$

The polynomial $x^{2}-a_{p}x+p$ can be interpreted as the characteristic
polynomial of the $p$-power Frobenius morphism on the the Tate-module
$T_{\ell}(E)$, where $\ell\neq p$ is a prime (and so $\alpha_{p}$ and
$\beta_{p}$ can be viewed as the corresponding eigenvalues). By the Riemann
Hypothesis for elliptic curves over finite fields (see for example [Sil09,
Theorem 2.3.1]), we have that

$|\alpha_{p}|=|\beta_{p}|=\sqrt{p}.$ (2.1)

If $p|N_{E}$, it is a standard fact that

$a_{p}\in\\{-1,0,1\\}.$ (2.2)

######  Lemma 2.1.

For all $s\in\mathbb{C}$ with $\mathrm{Re}(s)>3/2$, we have that

$-\frac{L^{\prime}(E,s)}{L(E,s)}=\sum_{n=1}^{\infty}\frac{b_{n}}{n^{s}},$

where

$b_{n}=\begin{cases}(\alpha_{p}^{k}+\beta_{p}^{k})\cdot\log p&\textrm{ if
}n=p^{k}\textrm{ and }p\nmid N_{E}\\\ a_{p}^{k}\cdot\log p&\textrm{ if
}n=p^{k}\textrm{ and }p|N_{E}\\\ 0&\textrm{otherwise.}\end{cases}$

###### Proof.

This follows by a direct computation, see for example [Spi15, Lemma 5.1.4] ∎

######  Theorem 2.2.

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

$\sum_{n\leq
x}\frac{b_{n}}{n^{s}}=-r\cdot\frac{x^{1-s}}{1-s}-\frac{L^{\prime}(E,s)}{L(E,s)}-\sum_{\rho\neq
1}\frac{x^{\rho-s}}{\rho-s}+\sum_{k=0}^{\infty}\frac{x^{-k-s}}{k+s},$

where $r=\textrm{ord}_{s=1}L(E,s)$, the sum over $\rho$ is taken over all non-
trivial zeros of $L(E,s)$ (excluding $\rho=1$) 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.

This is a standard argument to prove explicit formulas in analytic number
theory. By using Perron’s formula and the previous lemma, we have that

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

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 $s+z=1$,
$\mathrm{Res}_{z=1-s}\left(\frac{L^{\prime}(E,s+z)}{L(E,s+z)}\right)=r$, so
$\mathrm{Res}_{z=1-s}\left(-\frac{L^{\prime}(E,s+z)}{L(E,s+z)}\frac{x^{z}}{z}\right)=-r\cdot\frac{x^{1-s}}{1-s}$.
When $z=0$,
$\mathrm{Res}_{z=0}\left(-\frac{L^{\prime}(E,s+z)}{L(E,s+z)}\frac{x^{z}}{z}\right)=-\frac{L^{\prime}(E,s)}{L(E,s)}$.
If $\rho\neq 1$ is a non-trivial zero of $L(E,s)$,
$\mathrm{Res}_{z=\rho-s}\left(-\frac{L^{\prime}(E,s+z)}{L(E,s+z)}\frac{x^{z}}{z}\right)=-r_{\rho}\cdot\frac{x^{\rho-s}}{\rho-s}$,
where $r_{\rho}$ is the order of the zero $\rho$. Thus, we obtain a total
contribution of $-\sum_{\rho\neq 1}\frac{x^{\rho-s}}{\rho-s}$ counting
multiplicity for all the non-trivial zeros of $L(E,s)$ . Finally, the trivial
zeros of $L(E,s)$ are at $s=0,-1,-2,\ldots$ (which are all simple). Thus,
since
$\mathrm{Res}_{z=-k-s}\left(-\frac{L^{\prime}(E,s+z)}{L(E,s+z)}\frac{x^{z}}{z}\right)=\frac{x^{-k-s}}{k+s}$
for all $k\geq 0$, we get a total contribution of
$\sum_{k=0}^{\infty}\frac{x^{-k-s}}{k+s}$. ∎

To prove our main theorem, we will first relate the partial Euler product

$\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\cdot\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\prod_{\begin{subarray}{c}p\leq
x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\cdot\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-\alpha_{p}p^{-s})^{-1}(1-\beta_{p}p^{-s})^{-1}$

to the series $\sum\limits_{n\leq x}\frac{b_{n}}{n^{s}}$. We begin by noting
that

$\displaystyle\frac{d}{ds}\left(\log\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\prod_{\begin{subarray}{c}p\leq x\\\
p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}\right)=-\left(\text{I}(x)+\text{II}(x)+\text{III}(x)\right),$

where

$\text{I}(x)=\sum_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}\frac{\log p\cdot a_{p}}{p^{s}-a_{p}},\hskip
5.69054pt\text{II}(x)=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log p\cdot\alpha_{p}}{p^{s}-\alpha_{p}},\hskip
5.69054pt\text{III}(x)=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log p\cdot\beta_{p}}{p^{s}-\beta_{p}}.$

We now fix $s\in\mathbb{C}$ such that $\textrm{Re}(s)>1$. We write

$\text{I}(x)=\sum_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}\frac{\log p\cdot
a_{p}}{p^{s}}+\sum_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}\frac{\log p\cdot a_{p}^{2}}{p^{2s}}+\sum_{k\geq
3}\sum_{\begin{subarray}{c}p\leq x\\\ p|N_{E}\end{subarray}}\frac{\log p\cdot
a_{p}^{k}}{p^{ks}}$

$\text{II}(x)=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log
p\cdot\alpha_{p}}{p^{s}}+\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log p\cdot\alpha_{p}^{2}}{p^{2s}}+\sum_{k\geq
3}\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid N_{E}\end{subarray}}\frac{\log
p\cdot\alpha_{p}^{k}}{p^{ks}}$

$\text{III}(x)=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log
p\cdot\beta_{p}}{p^{s}}+\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log p\cdot\beta_{p}^{2}}{p^{2s}}+\sum_{k\geq
3}\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid N_{E}\end{subarray}}\frac{\log
p\cdot\beta_{p}^{k}}{p^{ks}}.$

On the other hand, using the definition of $b_{n}$ as given in Lemma 2.1,

$\sum_{n\leq x}\frac{b_{n}}{n^{s}}={\sum_{\begin{subarray}{c}p^{k}\leq x\\\
p|N_{E}\end{subarray}}}\frac{a_{p}^{k}\cdot\log
p}{p^{ks}}+{\sum_{\begin{subarray}{c}p^{k}\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{(\alpha_{p}^{k}+\beta_{p}^{k})\cdot\log
p}{p^{ks}}.$

We can write the first term as

$\displaystyle{\sum_{\begin{subarray}{c}p^{k}\leq x\\\
p|N_{E}\end{subarray}}}\frac{a_{p}^{k}\cdot\log
p}{p^{ks}}={\sum_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}}\frac{\log p\cdot a_{p}}{p^{s}}+\sum_{k\geq
2}{\sum_{\begin{subarray}{c}p\leq x^{1/k}\\\ p|N_{E}\end{subarray}}}\frac{\log
p\cdot a_{p}^{k}}{p^{ks}}$

and the second term as

$\displaystyle{\sum_{\begin{subarray}{c}p^{k}\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{ks}}$
$\displaystyle={\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}+\beta_{p})}{p^{s}}+{\sum_{\begin{subarray}{c}p\leq\sqrt{x}\\\
p\nmid N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}+\sum_{k\geq
3}{\sum_{\begin{subarray}{c}p\leq x^{1/k}\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{ks}}.$

Combining these observations it follows that

$\text{I}(x)+\text{II}(x)+\text{III}(x)=\sum_{n\leq
x}\frac{b_{n}}{n^{s}}+\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}+\sum_{k\geq
3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{ks}}+\sum_{k\geq
2}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq k\\\
p|N_{E}\end{subarray}}}\frac{\log p\cdot a_{p}^{k}}{p^{ks}}.$

Thus, in summary,

$\frac{d}{ds}\left(\log\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\prod_{\begin{subarray}{c}p\leq x\\\
p\nmid N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}\right)\newline =-\
\sum_{n\leq x}\frac{b_{n}}{n^{s}}-\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\
p\nmid N_{E}\end{subarray}}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}-\sum_{k\geq
3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{ks}}-\sum_{k\geq
2}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\
p|N_{E}\end{subarray}}}\frac{\log p\cdot a_{p}^{k}}{p^{ks}}.$ (2.3)

We can now prove our main theorem.

######  Theorem 2.3.

Assume the Riemann hypothesis holds for $L(E,s)$. Then for a complex number
$s\in\mathbb{C}$ with $1<\mathrm{Re}(s)<\frac{3}{2}$, we have that

$\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\cdot\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=L(E,s)\exp\left(-r\cdot\mathrm{Li}(x^{1-s})-R_{s}(x)+O\left(\frac{x^{2-2\mathrm{Re}(s)}-x^{1-\mathrm{Re}(s)}}{(2-2\mathrm{Re}(s))\log
x}\right)\right)\\\ $

where $r=\mathop{\mathrm{ord}}_{s=1}L(E,s)$, $\mathrm{Li}(x)$ is the principal
value of $\int\limits_{0}^{x}\frac{dt}{\log t}$, and $R_{s}(x)=\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\frac{x^{\rho-s}}{\rho-s}+\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz$.
Here, the sums are taken over all non-trivial zeros $\rho=1+i\gamma$ of
$L(E,s)$ (excluding $\rho=1$) 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 $1<\mathrm{Re}(s)<\frac{3}{2}$, we have by
Equation $\eqref{derivativeeqn}$ and Theorem 2.2 that

$\displaystyle\frac{d}{ds}$
$\displaystyle\left(\log\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s})^{-1}\prod_{\begin{subarray}{c}p\leq x\\\
p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}\right)=r\cdot\frac{x^{1-s}}{1-s}+\frac{L^{\prime}(E,s)}{L(E,s)}+\sum_{\rho\neq
1}\frac{x^{\rho-s}}{\rho-s}$ $\displaystyle-$
$\displaystyle\sum_{k=0}^{\infty}\frac{x^{-k-s}}{k+s}-\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\\\ p\nmid N_{E}\end{subarray}}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}-\sum_{k\geq
3}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{ks}}-\sum_{k\geq
2}{\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\
p|N_{E}\end{subarray}}}\frac{\log p\cdot a_{p}^{k}}{p^{ks}}.$

We now fix $s_{0}\in\mathbb{C}$ with $1<\mathrm{Re}(s)<\frac{3}{2}$.
Integrating this equation along the horizontal straight line from $s_{0}$ to
$\infty$, it follows that

$\displaystyle\log\prod_{\begin{subarray}{c}p\leq x\\\
p|N_{E}\end{subarray}}(1-a_{p}p^{-s_{0}})^{-1}\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s_{0}}+p^{1-2s_{0}})^{-1}=-r\cdot\int_{s_{0}}^{\infty}\frac{x^{1-s}}{1-s}ds-\int_{s_{0}}^{\infty}\frac{L^{\prime}(E,s)}{L(E,s)}ds$
$\displaystyle-\sum_{\rho\neq
1}\int_{s_{0}}^{\infty}\frac{x^{\rho-s}}{\rho-s}ds+\sum_{k=0}^{\infty}\int_{s_{0}}^{\infty}\frac{x^{-k-s}}{k+s}ds+\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\\\ p\nmid N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}ds$ $\displaystyle+\sum_{k\geq
3}\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{2s}}ds+\sum_{k\geq
2}\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\
p|N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log p\cdot
a_{p}^{k}}{p^{2s}}ds.$ (2.4)

We now analyze each of these integrals in turn.

  * •

We write $s_{0}=\sigma_{0}+it_{0}$ and parameterise the contour from $s_{0}$
to $\infty$ by the path $\gamma:[\sigma_{0},\infty]\rightarrow\mathbb{C}$
defined by $\gamma(y)=y+it_{0}$. Then

$\displaystyle\int_{s_{0}}^{\infty}\frac{x^{1-s}}{1-s}ds$ $\displaystyle=\log
x\int_{\sigma_{0}}^{\infty}\frac{e^{(1-(y+it_{0}))\log x}}{(1-(y+it_{0}))\log
x}dy=\int_{-\infty}^{(1-s_{0})\log x}\frac{e^{v}}{v}dv.$

By parameterzing this last integral in a similar manner, we conclude that

$\int_{s_{0}}^{\infty}\frac{x^{1-s}}{1-s}ds=\int_{0}^{e^{(1-s_{0})\log
x}}\frac{dt}{\log t}=\textrm{Li}(x^{1-s_{0}}).\\\ $

  * •

We have that

$\displaystyle\int_{s_{0}}^{\infty}\frac{L^{\prime}(E,s)}{L(E,s)}ds=-\log
L(E,s_{0}).$

  * •

We have that

$\displaystyle\sum_{\rho\neq
1}\int_{s_{0}}^{\infty}\frac{x^{\rho-s}}{\rho-s}ds$
$\displaystyle=\sum_{\rho\neq 1}\left(\frac{x^{\rho-s}}{-\log
x(\rho-s)}\Biggr{|}_{s_{0}}^{\infty}+\frac{1}{\log
x}\int_{s_{0}}^{\infty}\frac{x^{\rho-s}}{(\rho-s)^{2}}ds\right)$
$\displaystyle=\frac{1}{\log x}\sum_{\rho\neq 1}\frac{x^{\rho-s_{0}}}{\rho-
s_{0}}+\frac{1}{\log x}\sum_{\rho\neq
1}\int_{s_{0}}^{\infty}\frac{x^{\rho-s}}{(\rho-s)^{2}}ds$
$\displaystyle=R_{s_{0}}(x)\textrm{ by definition. }$

  * •

Note that

$\sum_{k=0}^{\infty}\frac{x^{-k-s}}{k+s}\ll\frac{1}{x^{\mathrm{Re}(s)}}\sum_{k=0}^{\infty}\frac{x^{-k}}{k+1}\ll\frac{1}{x^{\mathrm{Re}(s)}}.$

Thus,

$\int_{s_{0}}^{\infty}\sum_{k=0}^{\infty}\frac{x^{-k-s}}{k+s}\ll\int_{s_{0}}^{\infty}\frac{d|s|}{x^{\mathrm{Re}(s)}}=\frac{1}{x^{\mathrm{Re}(s_{0})}\log
x}.$

  * •

Using Equation 2.1, we have that

$\displaystyle\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s}}ds$
$\displaystyle=\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{(\alpha_{p}^{2}+\beta_{p}^{2})}{2p^{2s_{0}}}=O\left(\sum_{\begin{subarray}{c}\sqrt{x}<p\leq
x\\\ p\nmid N_{E}\end{subarray}}\frac{1}{p^{2\mathrm{Re}(s_{0})-1}}\right).$

To estimate this sum, note that

$\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{1}{p^{2\mathrm{Re}(s_{0})-1}}\ll\frac{1}{\log
x}\sum_{\sqrt{x}<p\leq x}\frac{\log p}{p^{2\mathrm{Re}(s_{0})-1}}.$

We can break this summation range into dyadic intervals of the form $[P,2P]$,
where $P$ is a prime. Then, using Chebyshev’s estimate we have that

$\sum_{P\leq p\leq 2P}\frac{\log
p}{p^{2\mathrm{Re}(s_{0})-1}}\ll\frac{1}{P^{2\mathrm{Re}(s_{0})-1}}\sum_{P\leq
p\leq 2P}\log
p\ll\frac{P}{P^{2\mathrm{Re}(s_{0})-1}}\ll\int_{P}^{2P}\frac{dt}{t^{2\mathrm{Re}(s_{0})-1}}.$

Thus,

$\sum_{\sqrt{x}\leq p\leq x}\frac{\log
p}{p^{2\mathrm{Re}(s_{0})-1}}\ll\int_{\sqrt{x}}^{x}\frac{dt}{t^{2\mathrm{Re}(s_{0})-1}}=\frac{x^{2-2\mathrm{Re}(s_{0})}-x^{1-\mathrm{Re}(s_{0})}}{2-2\mathrm{Re}(s_{0})}.$

We hence conclude that

$\sum_{\begin{subarray}{c}\sqrt{x}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{p}^{2}+\beta_{p}^{2})}{p^{2s_{0}}}ds=O\left(\frac{x^{2-2\mathrm{Re}(s_{0})}-x^{1-\mathrm{Re}(s_{0})}}{(2-2\mathrm{Re}(s_{0}))\log
x}\right).$

Note that this error term subsumes the error term in the previous bullet
point.

  * •

By the same reasoning, and noting that $\ll\log x$ terms contribute to the
first sum, we have that

$\sum_{k\geq 3}\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\ p\nmid
N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log
p\cdot(\alpha_{p}^{k}+\beta_{p}^{k})}{p^{3s_{0}}}ds=O\left(\frac{x^{3-3\mathrm{Re}(s_{0})}-x^{1-\mathrm{Re}(s_{0})}}{(3-3\mathrm{Re}(s_{0}))\log
x}\cdot\log x\right).$

and so this error term can be subsumed into the previous error term. Finally,
since $a_{p}\in\\{-1,0,1\\}$, the contribution from the term

$\sum_{k\geq 2}\sum_{\begin{subarray}{c}x^{1/k}<p\leq x\\\
p|N_{E}\end{subarray}}\int_{s_{0}}^{\infty}\frac{\log p\cdot
a_{p}^{k}}{p^{2s}}ds$

can certainly also be absorbed into the previous error term.

Inserting all of the above estimates into Equation (2) and exponentiating
proves the result. ∎

##  3\. Refinement of the error term for $\psi_{E}(x)$

Let $\psi_{E}(x)=\sum\limits_{\begin{subarray}{c}p^{k}\leq x\\\ p\nmid
N_{E}\end{subarray}}(\alpha_{p}^{k}+\beta_{p}^{k})\log p$. The Riemann
Hypothesis for $L(E,s)$ is equivalent to the fact that $\psi_{E}(x)=O(x(\log
x)^{2})$. In this section, conditional on the Riemann Hypothesis for $L(E,s)$,
we use a method due to Gallagher [Gal80] to improve the error term outside a
set of finite logarithmic measure. To do so, we will need a general result
(see for instance [Sel92]) on the vertical distribution of zeros for a wide
class of $L$-functions. For our elliptic curve $L$-function, the result is the
following (see also [KM23, Theorem 2]).

######  Theorem 3.1.

Let $N_{E}(t)$ be the number of zeros $\rho=\beta+i\gamma$ of $L(E,s)$
satisfying $0<\gamma<t$. Then

$N_{E}(t)=\frac{\alpha_{E}}{\pi}t(\log t+c)+O(\log t),$

where $c$ is a constant and $\alpha_{E}$ is a constant depending on $E$.

######  Corollary 3.2.

We have that

$\sum_{\rho}\frac{1}{|\rho|^{2}}$

converges, where the sum is taken over all non-trivial zeros of $L(E,s)$.

###### Proof.

Note that

$\sum_{\rho}\frac{1}{|\rho|^{2}}=\sum_{t=0}^{\infty}\sum_{t<|\gamma|\leq
t+1}\frac{1}{|\rho|^{2}}.$

By Theorem 3.1, the number of terms $t<\gamma\leq t+1$ is $\ll\log t$. Thus,

$\sum_{\rho}\frac{1}{|\rho|^{2}}\ll\sum_{t=1}^{\infty}\frac{\log
t}{t^{2}}<\infty.$

∎

We will also need the following estimate on mean values of exponential sums.

######  Lemma 3.3.

Let $\mathcal{A}$ be a discrete subset of $\mathbb{R}$. For each
$\nu\in\mathcal{A}$, let $c(\nu)$ be a complex number such that the series

$S(u)=\sum_{\nu\in\mathcal{A}}c(\nu)e^{2\pi i\nu u}$

is absolutely convergent. Then for any $\theta\in(0,1)$, we have that

$\int_{-U}^{U}|S(u)|^{2}du\leq\left(\frac{\pi\theta}{\sin(\pi\theta)}\right)^{2}\int_{-\infty}^{\infty}\left|\frac{U}{\theta}\sum_{t\leq\nu\leq
t+\frac{\theta}{U}}c(\nu)\right|^{2}dt.$

###### Proof.

This lemma was originally proven by Gallagher (see [Gal70, Lemma 1]), but the
version stated above is [Koy16, Lemma 1]. ∎

######  Theorem 3.4.

Assume the Riemann Hypothesis for $L(E,s)$. Then there exists a subset
$S\subseteq\mathbb{R}_{\geq 2}$ of finite logarithmic measure such that for
all $x\not\in S$,

$\psi_{E}(x)=O(x(\log\log x)^{2}).$

###### Proof.

Let $X$ be such that $X\leq x\ll X$. By the Riemann von-Mangoldt explicit
formula for $L(E,s)$ (see [Fio14, Lemma 2.1] and [Rub13, Equation 1.10]), we
have that

$\psi_{E}(x)=-\sum_{\begin{subarray}{c}\rho\\\ |\gamma|\leq
X\end{subarray}}\frac{x^{\rho}}{\rho}+O(\log^{2}X).$

By Theorem 3.1, the number of terms $t<\gamma\leq t+1$ is $\ll\log t$. Thus,

$\sum_{t<|\gamma|\leq t+1}\frac{x^{\rho}}{\rho}\ll x\cdot\frac{\log t}{t}$

and so

$\sum_{0<|\gamma|<\log X}\frac{x^{\rho}}{\rho}\ll X\sum_{j=1}^{\lfloor\log
X\rfloor}\frac{\log(j+1)}{j}dt\ll X\int_{1}^{\log X}\frac{\log
t}{t}dt=\frac{X(\log\log X)^{2}}{2}\ll x(\log\log x)^{2}.$ (3.1)

Let $T\leq X$. We now consider the integral

$\int_{X}^{eX}\left|\sum_{T<\gamma\leq
X}\frac{x^{\rho}}{\rho}\right|^{2}\frac{dx}{x^{3}}.$

By making the substitution $x=e^{u}$ and then $u=\frac{1}{2}+\log X+2\pi t$,
this integral equals

$\int_{\log X}^{1+\log X}\left|\sum_{T<\gamma<X}\frac{e^{i\gamma
u}}{\rho}\right|^{2}du=2\pi\cdot\int_{-\frac{1}{4\pi}}^{\frac{1}{4\pi}}\left|\sum_{T<\gamma<X}\frac{e^{i\gamma(\frac{1}{2}+\log
X)}}{\rho}\cdot e^{2\pi i\gamma t}\right|^{2}dt.$

Setting $\mathcal{A}=\\{\gamma:T<\gamma\leq X\\}$, $\theta=\frac{1}{4\pi}$,
and

$c(\gamma)=\begin{cases}\frac{e^{i\gamma(\frac{1}{2}+\log X)}}{\rho}&\text{if
}T<\gamma\leq X\\\ 0&\text{otherwise}\end{cases}$

we can apply Lemma 3.3 conclude that

$\int_{X}^{eX}\left|\sum_{T<\gamma\leq
X}\frac{x^{\rho}}{\rho}\right|^{2}\frac{dx}{x^{3}}\ll\int_{-\infty}^{\infty}\left(\sum_{\begin{subarray}{c}T<\gamma\leq
X\\\ t\leq\gamma\leq
t+1\end{subarray}}\frac{1}{|\rho|}\right)^{2}dt\ll\int_{T-1}^{X}\left(\sum_{t\leq\gamma\leq
t+1}\frac{1}{|\rho|}\right)^{2}dt,$

Using again the fact that the number of terms $t<\gamma\leq t+1$ is $\ll\log
t$ we get that

$\displaystyle\int_{T-1}^{X}\left(\sum_{t<\gamma<t+1}\frac{1}{|\rho|}\right)^{2}dt$
$\displaystyle\ll\int_{T-1}^{X}\left(\frac{\log t}{t}\right)^{2}dt$
$\displaystyle=\frac{-\log^{2}t-2\log t-2}{t}\Biggr{|}_{T-1}^{X}$
$\displaystyle\ll\frac{\log^{2}T}{T}.$

Define a set $S_{T}$ by setting

$S_{T}=\\{x\in[e^{T},e^{T+1}]:\left|\sum_{T<|\gamma|\leq
e^{T}}\frac{x^{\rho}}{\rho}\right|\geq x(\log\log x)^{2}\\}.$

Setting $X=e^{T}$, we conclude from above that

$\displaystyle\frac{\log^{2}T}{T}$
$\displaystyle\gg\int_{S_{T}}\left|\sum_{T<\gamma\leq
X}\frac{x^{\rho}}{\rho}\right|^{2}\frac{dx}{x^{3}}$
$\displaystyle\gg\int_{S_{T}}\frac{(\log\log x)^{4}}{x}dx$
$\displaystyle\gg(\log\log e^{T})^{4}\int_{S_{T}}\frac{dx}{x}$
$\displaystyle=(\log T)^{4}\cdot\mu^{\times}(S_{T}).$

Thus,

$\mu^{\times}(S_{T})\ll\frac{1}{T\log^{2}T}.$

Define the set

$S=\bigcup_{T=2}^{\infty}S_{T}.$

Then

$\mu^{\times}(S)\ll\sum_{T=2}^{\infty}\frac{1}{T\log^{2}T}<\infty.$

Thus, $S$ has finite logarithmic measure and so, by definition of $S$ and by
Equation (3.1), we conclude that

$\psi_{E}(x)=O(x(\log\log x)^{2})$

for all $x\not\in S$. ∎

##  4\. Applications

######  Theorem 4.1.

Assume the Riemann Hypothesis for $L(E,s)$. Then there exists a subset
$S\subseteq\mathbb{R}_{\geq 2}$ of finite logarithmic measure such that for
all $x\not\in S$,

$\prod_{p\leq x}\frac{N_{p}}{p}\asymp(\log x)^{r},$

where $r=\mathop{\mathrm{ord}}_{s=1}L(E,s)$.

###### Proof.

The idea of the proof is to set $s=1+\frac{1}{x}$ and estimate the left-hand
and right-hand sides of Theorem 2.3 as $x\to\infty$. When $s=1+\frac{1}{x}$,
the left-hand side of Theorem 2.3 (ignoring the finitely many terms for primes
which divide the conductor) equals

$\displaystyle\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}$
$\displaystyle=\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-1-1/x}+p^{-1-2/x})^{-1}$
$\displaystyle=\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\left(1-a_{p}p^{-1}\left(1+O\left(\frac{\log
p}{x}\right)\right)+p^{-1}\left(1+O\left(\frac{\log
p}{x}\right)\right)\right)^{-1}$
$\displaystyle=\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-1}+p^{-1})^{-1}\cdot\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid N_{E}\end{subarray}}\left(1+O\left(\frac{a_{p}\cdot\log
p}{px}\right)\right),$

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

$\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\left(1+O\left(\frac{a_{p}\cdot\log
p}{px}\right)\right)=1+O\left(\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\frac{|a_{p}|\cdot\log p}{px}\right)=1+O\left(\frac{\log
x}{\sqrt{x}}\right)=1+o(1),$

where we used the fact that $|a_{p}|\leq 2\sqrt{p}$ and the classical estimate
$\sum_{p\leq x}\frac{\log p}{p}=\log x+O(1)$. Thus, when $s=1+\frac{1}{x}$,

$\prod_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-s}+p^{1-2s})^{-1}\sim\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid
N_{E}\end{subarray}}(1-a_{p}p^{-1}+p^{-1})^{-1}=\prod_{\begin{subarray}{c}p\leq
x\\\ p\nmid N_{E}\end{subarray}}\frac{p}{N_{p}}.$ (4.1)

We now estimate the right-hand side of Theorem 2.3. Write

$L(E,s)=a_{r}(s-1)^{r}+a_{r+1}(s-1)^{r+1}+\cdots$

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

To estimate the contribution coming from the term $\mathrm{Li}(x^{1-s})$, 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 $s=1+\frac{1}{x}$ yields that

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

To estimate the contribution coming from the term $R_{s}(x)=\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\frac{x^{\rho-s}}{\rho-s}+\frac{1}{\log
x}\sum\limits_{\rho\neq 1}\int_{s}^{\infty}\frac{x^{\rho-z}}{(\rho-z)^{2}}dz$
when $s=1+\frac{1}{x}$, we begin by noting that $\sum_{\rho\neq
1}\frac{1}{\rho-s}-\sum_{\rho\neq 1}\frac{1}{\rho}=\sum_{\rho\neq
1}\frac{s}{(\rho-s)\rho}$ and, using Corollary 3.2, we have that
$\sum_{\rho\neq
1}\frac{s}{(\rho-s)\rho}\ll\sum_{\rho}\frac{1}{|\rho|^{2}}<\infty$. Thus,

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

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

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

Using the Riemann von-Mangoldt explicit formula for $L(E,s)$ (see [Rub13,
Equation 1.10])

$\psi_{E}(x)=-\sum_{\rho}\frac{x^{\rho}}{\rho}+o(x^{1/2}),$

it follows that

$\sum_{\rho\neq 1}\frac{x^{\rho-s}}{\rho-s}\ll\frac{1}{x}\psi_{E}(x)+O(1).$

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

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

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

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

To estimate the second quantity in the definition of $R_{s}(x)$, we note that
since $\sum\limits_{\rho}\frac{1}{|\rho|^{2}}$ converges by Corollary 3.2,

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

The final term
$O\left(\frac{x^{2-2\mathrm{Re}(s)}-x^{1-\mathrm{Re}(s)}}{(2-2\mathrm{Re}(s))\log
x}\right)$ equals $O(1)$ when $s=1+\frac{1}{x}$, as can be seen by Taylor
expanding the numerator. Thus, in summary, when $s=1+1/x$, we conclude that
for all $x\not\in S$,

$\displaystyle
L(E,s)\exp\left(-r\cdot\mathrm{Li}(x^{1-s})-R_{s}(x)+O\left(\frac{x^{2-2\mathrm{Re}(s)}-x^{1-\mathrm{Re}(s)}}{(2-2\mathrm{Re}(s))\log
x}\right)\right)$ $\displaystyle=L(E,s)\exp\left(-r\log\left(\frac{\log
x}{x}\right)+O(1)\right)$
$\displaystyle\asymp\frac{a_{r}}{x^{r}}\exp\left(-r\log\left(\frac{\log
x}{x}\right)\right)$ $\displaystyle\asymp\frac{1}{(\log x)^{r}}.$

Combining this with equation (4.1) proves the theorem. ∎

######  Corollary 4.2.

Let $E/\mathbb{Q}$ be an elliptic curve. Suppose that

$\prod_{p\leq x}\frac{N_{p}}{p}\sim C(\log x)^{\mathrm{rank}(E(\mathbb{Q}))}.$

Then

$\mathop{\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}(E(\mathbb{Q})).$

###### Proof.

Assuming the Riemann Hypothesis for $L(E,s)$, this follows directly from the
previous theorem. It is known that (see [Gol82, Theorem 2(a)] or [KM05,
pp.334]) that the asymptotic $\prod_{p\leq x}\frac{N_{p}}{p}\sim C(\log
x)^{\mathrm{rank}(E(\mathbb{Q}))}$ implies the Riemann Hypothesis for
$L(E,s)$, so the corollary is true unconditionally. For completeness, we
explain the proof of this implication here. Note that

$\prod_{p\leq
x}\frac{N_{p}}{p}=\prod_{p|N_{E}}\left(1-\frac{a_{p}}{p}\right)\prod_{p\nmid
N_{E}}\left(1-\frac{\alpha_{p}}{p}\right)\left(1-\frac{\beta_{p}}{p}\right).$

Taking logarithms on both sides yields and computing the remainder term (see
[KM05, Lemma 1] for details) yields that

$\log\prod_{p\leq x}\frac{N_{p}}{p}=\sum_{n\leq x}c_{n}+O(1),$

where

$c_{n}=\begin{cases}-\frac{(\alpha_{p}^{k}+\beta_{p}^{k})}{kp^{k}}&\textrm{ if
}n=p^{k}\textrm{ and }p\nmid N_{E}\\\ -\frac{a_{p}^{k}}{kp^{k}}&\textrm{ if
}n=p^{k}\textrm{ and }p|N_{E}\\\ 0&\textrm{otherwise.}\end{cases}$

Thus, by hypothesis, $\sum_{n\leq
x}c_{n}=O(\mathrm{rank}(E(\mathbb{Q}))\cdot\log\log x)$ and partial summation
yields that

$\log
L(E,s+1)=-\sum_{n=1}^{\infty}\frac{c_{n}}{n^{s}}=-s\int_{1}^{\infty}(\sum_{n\leq
x}c_{n})x^{-(s+1)}dx\textrm{ for all }s\in\mathbb{C}\textrm{ with
}\mathrm{Re}(s)>1/2.$

Hence, $\log L(E,s+1)$ is holomorphic for all $s\in\mathbb{C}$ with
$\mathrm{Re}(s)>0$ and so $L(E,s)\neq 0$ for all $s\in\mathbb{C}$ with
$\mathrm{Re}(s)>1$. ∎

## References

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∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: Medical Imaging Techniques, Simav Vocational School of Health
Services, Kutahya Health Sciences University, TR- 43500 Kutahya, Turkey

# Relativistic Landau levels for a fermion-antifermion pair interacting
through Dirac oscillator interaction

Abdullah Guvendie1,addr1

(Received: date / Accepted: date)

###### Abstract

We introduce a unique model for a fermion antifermion pair interacting through
Dirac oscillator interaction in the presence of external uniform magnetic
field. In order to acquire a non perturbative energy spectrum for such a
system we solve the corresponding form of a fully covariant two body Dirac
equation. The dynamic symmetry of the system allows to study in three
dimensions and the corresponding equation leads a $4\times 4$ dimensional
matrix equation for such a static and spinless composite system. We can obtain
an exact solution of the matrix equation and arrive at a spectrum (in closed
form) in energy domain. The obtained energy spectrum shows that the composite
system that under scrutiny behaves like a single relativistic quantum
oscillator. As a result, we obtain relativistic Landau levels of a fermion
antifermion pair interacting through Dirac oscillator interaction and
determine the components of the corresponding bi-spinor. We think that our
results can provide enlightening informations about the quark antiquark
systems and thus mass formula for mesons.

###### Keywords:

Dirac oscillator Relativistic Landau levels Two-body Dirac equation Magnetic
field

††journal: Eur. Phys. J. C

##  1 Introduction

The Dirac oscillator has been introduced as a new type of interaction in the
Dirac equation ito1967example ; cook1971relativistic ; moshinsky1989dirac
(also see nikolsky1930oszillatorproblem ). The corresponding Hamiltonian to
this equation is linear in both momentum and spatial coordinate ito1967example
; cook1971relativistic  and this instance of Dirac equation gives a simple
quantum oscillator solution with a strong spin-orbit coupling term in the non-
relativistic limit moshinsky1989dirac ; carvalho2011dirac . This is why it has
been called as Dirac oscillator moshinsky1989dirac ; book:1028629 . It has
been shown that the Dirac oscillator interaction corresponds to an anomalous
magnetic interaction when this interaction is written in the manifestly
covariant form in the Dirac equation bentez1990solution . Taking advantage the
gauge invariance of the electromagnetic interaction and selecting an
appropriate gauge, it has been shown that the Dirac oscillator interaction
occurs as if it were produced by an infinite sphere carrying a uniform charge-
density distribution, resulting in a linearly growing electric field
bentez1990solution . It has also been suggested that the Dirac oscillator
interaction can be regarded as an alternative confinement potential for heavy
quarks in quantum chromodynamics moreno1989covariance . Hence, we can think
that a well-established mathematical model for tight-binding fermion-fermion
systems may be useful to better understand the physics behind such systems
moshinsky1993barut ; franco2013first ; kulikov2007relativistic .

The Dirac oscillator interaction has been studied in many areas of physics
such as in mathematical physics zarrinkamar2010dirac ; bermudez2008chirality ;
de2006relating ; alhaidari2006dirac ; quesne2005dirac ; ho2004quasi ;
villalba1994exact , nuclear (or subnuclear) physics munarriz2012spectroscopy ;
wang2012zitterbewegung ; romera2011revivals  (also see
grineviciute2012relativistic ; faessler2005description ), quantum optics
bermudez2008chirality ; dodonov2002nonclassical ; bermudez2007mesoscopic ;
bermudez2007exact ; dutta2013pseudo  and molecular dynamics franco2013first .
One of the most important aspects of this system is that it has found many
application areas in various branches of physics. The Dirac oscillator system
one of the most useful tools in mathematical physics due to the fact it is an
exactly solvable model. Thus, a great majority of the studies based on the
determination of the influence of spacetime topology on the relativistic
dynamics of the mentioned quantum systems are based on this useful system
carvalho2011dirac ; bakke2013interaction ; salazar2019algebraic ;
bueno2014quantum ; oliveira2019topological ; hosseinpour2019dirac ;
bakke2013rotating . The Dirac oscillator has been applied in many other
physical context such as graphene sadurni2011dirac  and it has been used to
determine the quantum gravity effects benzair2017propagator ; stetsko20191+ .
However, it is worth to mention that the Dirac oscillator is normally utilized
in the context of many-body theory moshinsky1993barut .

A phenomenological way to describe the dynamics of many-body systems is to use
one-time equations including free Hamiltonians for each particle plus
interparticle interaction potentials kemmer1937 ; fermi1949 . Although, the
history of such equations goes long way back van1997 ; giachetti2019 , one can
see that there are still different relativistic two-body Dirac equations in
use today in the literature van1997 ; giachetti2019 ; breit1929 ; salpeter1951
; barut1985derivation ; guvendi2019exact . In this study, we deal with a
fully-covariant two-body Dirac equation moshinsky1993barut ; guvendi2019exact
; guvendi2020interacting ; guvendi2020relativistic ; barut1985 ; barut1987a ;
barut1987b  that derived from quantum electrodynamics with the help of action
principle barut1985derivation . This equation includes the most general
electric and magnetic potentials and leads a $16\times 16$ matrix equation in
$4$-dimensions barut1985 ; barut1987a ; barut1987b . Therefore, in
$4$-dimensions, the separation of the angular and radial variables requires to
be applied group theoretical methods barut1985 ; barut1987a ; barut1987b ;
aydin1988relativistic . It has been announced that the well-known energy
spectrum for one-electron atoms or fermion-antifermion systems can be obtained
via perturbative solution of the obtained radial equations moshinsky1993barut
; barut1985 ; barut1987a ; barut1987b . Recently, it has been shown that this
fully-covariant two-body Dirac equation can be exactly solvable for both low
dimensional systems guvendi2019exact  and for systems that have dynamical
symmetries guvendi2020relativistic ; guvendi2020interacting , without
considering any group theoretical method. This fully covariant two-body Dirac
equation can also be used to determine the relativistic dynamics of any
fermion-fermion system in curved spacetimes guvendi2020interacting . This
equation has been used to determine the effect of gravitational field produced
by cosmic strings on the energy of unstable composite systems such as
positronium and the results, in principle, have shown that an ortho-
positronium system have a potential to prove the existence of cosmic strings
in the universe guvendi2020interacting .

In this present manuscript, we introduce a unique presentation to determine
the relativistic Landau levels of a fermion-antifermion pair interacting
through Dirac oscillator interaction. In order to acquire this, we can write
the fully-covariant two-body Dirac equation for arbitrary two Dirac
oscillators interacting with an external uniform magnetic field. The system
that under scrutiny has translational symmetry. Therefore, we can investigate
the relativistic dynamics of such a composite system in $3$-dimensional flat
Minkowski spacetime background by choosing the interaction of each particle
with the external uniform magnetic field in the symmetric gauge. We can
separate, covariantly, center of mass motion coordinates and relative motion
coordinates and then arrive at a $4\times 4$ dimensional matrix equation,
resulting a set of coupled equations. For a static and spinless composite
system formed by a fermion-antifermion pair interacting through Dirac
oscillator interaction in the presence of external uniform magnetic field, the
coupled equations lead to an exactly solvable $2^{nd}$ order wave equation. As
a result, we can obtain a non-perturbative spectrum (in closed-form) in energy
domain. The obtained energy spectrum shows that the system we deal with
behaves like a relativistic quantum oscillator carrying total rest mass of
such a composite system. Among the previously announced studies in the
literature, one can find a few study that concentrate on the relativistic
dynamics of fermion-fermion systems with Dirac oscillator interaction (see
moshinsky1993barut ; moshinsky1991anomalous ; moshinsky1995supersymmetry ;
bednar1997connection ). We hope that our exact results can contribute to fill
this gap in the literature. Moreover, we think that our results can give
insight into the quark-antiquark systems.

##  2 Mathematical Model

We suggest that the relativistic dynamics of a fermion-antifermion pair
interacting through Dirac oscillator interaction in the presence of an
external uniform magnetic field can be investigated, without considering any
charge-charge interaction, by using the corresponding form of the following
fully-covariant two-body Dirac equation moshinsky1993barut ; guvendi2019exact
; guvendi2020interacting ; guvendi2020relativistic ; barut1985 ; barut1987a ;
barut1987b ,

$\displaystyle\left\\{H^{\left(1\right)}\otimes\gamma^{0^{\left(2\right)}}+\gamma^{0^{\left(1\right)}}\otimes
H^{\left(2\right)}\right\\}\chi\left(\textbf{x}_{1},\textbf{x}_{2}\right)=0,$
$\displaystyle\leavevmode\resizebox{422.77661pt}{}{$H^{\left(1\right)}=\left[\gamma^{\eta^{\left(1\right)}}\pi^{\left(1\right)}_{\eta}+ib_{1}\textbf{I}_{2}\right],\quad
H^{\left(2\right)}=\left[\gamma^{\eta^{\left(2\right)}}\pi^{\left(2\right)}_{\eta}+ib_{2}\textbf{I}_{2}\right]$},$
$\displaystyle\leavevmode\resizebox{195.12767pt}{}{$\pi^{\left(1\right)}_{\eta}=\left(\partial^{\left(1\right)}_{\eta}+i\frac{e_{1}A^{\left(1\right)}_{\eta}}{\hbar
c}-\Gamma_{\eta}^{\left(1\right)}\right),\quad\pi^{\left(2\right)}_{\eta}=\left(\partial^{\left(2\right)}_{\eta}+i\frac{e_{2}A^{\left(2\right)}_{\eta}}{\hbar
c}-\Gamma_{\eta}^{\left(2\right)}\right)$},$ $\displaystyle
b_{1}=\frac{m_{1}c}{\hbar},\quad
b_{2}=\frac{m_{2}c}{\hbar},\quad\left(\eta=0,1,2.\right),$
$\displaystyle\partial_{\eta}=(\partial_{0},\partial_{1},\partial_{2}.),$ (1)

here, the superscripts $(1)$ and $(2)$ refer to the first fermion with mass
$m_{1}$ and the second fermion with mass $m_{2}$, respectively,
$\textbf{I}_{2}$ is $2$-dimensional unit matrix,
$\Psi\left(\mathbf{x}_{1},\mathbf{x}_{2}\right)$ is the bi-local composite
field that is constructed by Kronocker production $(\otimes)$ of arbitrary
massive two Dirac fields as in the following,

$\displaystyle\chi\left(\mathbf{x}_{1},\mathbf{x}_{2}\right)=\chi_{1}\left(\mathbf{x}_{1}\right)\otimes\chi_{2}\left(\mathbf{x}_{2}\right),$
(2)

$\hbar$ is the reduced Planck constant, $c$ is the speed of light, $e_{1}$ and
$e_{2}$ represent the charges of these fermions, $\Gamma_{\eta}$ and
$A_{\eta}$ stand for the spinor connections and vector potentials,
respectively. Here, it is important to notice that, the $\gamma^{0}$ mean
$\gamma^{\eta}\lambda_{\eta}$ in every where in Eq. (1) and the
$\lambda_{\eta}$ is a timelike vector $\lambda_{\eta}=(1,0,0)$ barut1986center
. This equation includes spin algebra spanned by Kronocker productions of the
Dirac matrices. More details about this equation can be found in
barut1985derivation . Thanks to the dynamic symmetry of the system that under
scrutiny we can study in $3$-dimensional flat Minkowski spacetime background.
In terms of cartesian coordinates, this spacetime background is represented by
the following line element guvendi2019exact ; guvendi2020relativistic ,

$\displaystyle ds^{2}=c^{2}dt^{2}-(dx^{2}+dy^{2}).$ (3)

For the system that under scrutiny in this spacetime background, it is clear
that the spinor connections in Eq. (1) does not effect the dynamics, since
they vanish (see sucu2007 ). In the presence of the external uniform magnetic
field, the vector potentials in Eq. (1) can be chosen in the symmetric gauge
as follows guvendi2020relativistic ,

$\displaystyle A^{\left(1\right)}_{t}=0,\quad
A^{\left(1\right)}_{x}=-\frac{B_{0}y_{1}}{2},\quad
A^{\left(1\right)}_{y}=\frac{B_{0}x_{1}}{2},$ $\displaystyle
A^{\left(2\right)}_{t}=0,\quad
A^{\left(2\right)}_{x}=-\frac{B_{0}y_{2}}{2},\quad
A^{\left(2\right)}_{y}=\frac{B_{0}x_{2}}{2},$ (4)

in which the $B_{0}$ relates with the strength of external uniform magnetic
field and also the written spatial coordinate
pairs($x_{v},y_{v}\left(v=1,2.\right)$) correspond to the spatial coordinates
of the particles in the spacetime background given in Eq. (3). According to
the signature $(+,-,-)$ in Eq. (3) we can choose the Dirac matrices as in the
following guvendi2019exact ; guvendi2020relativistic ,

$\displaystyle\gamma^{t^{\left(1,2\right)}}=\sigma^{z},\quad\gamma^{x^{\left(1,2\right)}}=i\sigma^{x},\quad\gamma^{y^{\left(1,2\right)}}=i\sigma^{y},$
$\displaystyle\sigma^{x}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right),\quad\sigma^{y}=\left(\begin{array}[]{cc}0&-i\\\ i&0\\\
\end{array}\right),$ (9)
$\displaystyle\quad\sigma^{z}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right),$ (12)

where, $\sigma^{x}$, $\sigma^{y}$ and $\sigma^{z}$ are the Pauli spin
matrices. Also, we can introduce two dimensional two Dirac oscillators via the
non-minimal substitution terms as follows ito1967example ;
cook1971relativistic ; moshinsky1989dirac ,

$\displaystyle\leavevmode\resizebox{422.77661pt}{}{$\partial_{x}^{\left(1\right)}\rightarrow\partial_{x}^{\left(1\right)}+\frac{m_{1}\omega_{1}}{\hbar}\gamma^{0^{\left(1\right)}}x_{1},\quad\partial_{y}^{\left(1\right)}\rightarrow\partial_{y}^{\left(1\right)}+\frac{m_{1}\omega_{1}}{\hbar}\gamma^{0^{\left(1\right)}}y_{1}$},$
$\displaystyle\leavevmode\resizebox{422.77661pt}{}{$\partial_{x}^{\left(2\right)}\rightarrow\partial_{x}^{\left(2\right)}+\frac{m_{2}\omega_{2}}{\hbar}\gamma^{0^{\left(2\right)}}x_{2},\quad\partial_{y}^{\left(2\right)}\rightarrow\partial_{y}^{\left(2\right)}+\frac{m_{2}\omega_{2}}{\hbar}\gamma^{0^{\left(2\right)}}y_{2}$},$
(13)

in which $\omega_{1}$ is the frequency of the oscillator with mass $m_{1}$ and
$\omega_{2}$ is the frequency of the oscillator with mass $m_{2}$. Now, one
can separate the center of mass motion coordinates and relative motion
coordinates via the following expressions guvendi2019exact ,

$\displaystyle
R_{\eta}=\frac{1}{M}\left(m_{1}x_{\eta}^{\left(1\right)}+m_{2}x_{\eta}^{\left(2\right)}\right),\quad
r_{\eta}=x_{\eta}^{\left(1\right)}-x_{\eta}^{\left(2\right)},$ $\displaystyle
x_{\eta}^{\left(1\right)}=\frac{m_{2}}{M}r_{\eta}+R_{\eta},\quad
x_{\eta}^{\left(2\right)}=-\frac{m_{1}}{M}r_{\eta}+R_{\eta},$
$\displaystyle\partial_{x_{\eta}}^{\left(1\right)}=\partial_{r_{\eta}}+\frac{m_{1}}{M}\partial_{R_{\eta}},\quad\partial_{x_{\eta}}^{\left(2\right)}=-\partial_{r_{\eta}}+\frac{m_{2}}{M}\partial_{R_{\eta}},$
$\displaystyle\partial_{x_{\eta}}^{\left(1\right)}+\partial_{x_{\eta}}^{\left(2\right)}=\partial_{R_{\eta}}.$
(14)

##  3 Coupled equations for the components of bi-spinor

Provided that the interaction is time-independent and the momentum of the
center of mass is a constant of motion one can define the spinor $\chi$ as
follows,

$\displaystyle\chi\left(R_{0},\textbf{R},\textbf{r}\right)=e^{-iwt}e^{i\textbf{k}.\textbf{R}}\psi\left(\textbf{r}\right),$
$\displaystyle\psi\left(\textbf{r}\right)=\left(\begin{array}[]{c}\varphi_{1}\left(\textbf{r}\right)\\\
\varphi_{2}\left(\textbf{r}\right)\\\ \varphi_{3}\left(\textbf{r}\right)\\\
\varphi_{4}\left(\textbf{r}\right)\end{array}\right),$ (19)

here, $w$ is total frequency that determined according to the proper time of
the system ($R_{0}$) barut1985derivation , k relates with the center of mass
momentum as $\hbar\textbf{k}$, R is spatial position vector of the center of
mass and the r is spatial relative vector between the particles. Assuming the
center of mass is located at the origin of the spacetime background and does
not carry momentum ($\hbar\textbf{k}=0$) for oppositely charged
($e_{1}=e,e_{2}=-e$)111This case requires that
$\omega_{1}=\omega,\omega_{2}=-\omega$ moshinsky1993barut . and equal massive
($m_{1}=m_{2}=m$) two Dirac oscillators in the presence of an external uniform
magnetic field, we obtain the following matrix equation via the Eq. (4), Eq.
(12), Eq. (13), Eq. (14), Eq. (19) and Eq. (1),

$\displaystyle-\left(\gamma^{t}\otimes\gamma^{t}\right)i\varepsilon\psi+i\frac{mc}{\hbar}\left(\textbf{I}_{2}\otimes\gamma^{t}+\gamma^{t}\otimes\textbf{I}_{2}\right)\psi$
$\displaystyle+\left(\gamma^{x}\otimes\gamma^{t}-\gamma^{t}\otimes\gamma^{x}\right)\partial_{x}\psi$
$\displaystyle+\left(\gamma^{y}\otimes\gamma^{t}-\gamma^{t}\otimes\gamma^{y}\right)\partial_{y}\psi$
$\displaystyle+i\kappa
x\left(\gamma^{y}\otimes\gamma^{t}+\gamma^{t}\otimes\gamma^{y}\right)-i\kappa
y\left(\gamma^{x}\otimes\gamma^{t}+\gamma^{t}\otimes\gamma^{x}\right)\psi$
$\displaystyle+\xi
x\left(\left(\gamma^{x}\gamma^{t}\right)\otimes\gamma^{t}+\gamma^{t}\otimes\left(\gamma^{x}\gamma^{t}\right)\right)\psi$
$\displaystyle+\xi
y\left(\left(\gamma^{y}\gamma^{t}\right)\otimes\gamma^{t}+\gamma^{t}\otimes\left(\gamma^{y}\gamma^{t}\right)\right)\psi=0,$
$\displaystyle\varepsilon=\frac{w}{c},\quad\kappa=\frac{eB_{0}}{2\hbar
c},\quad\xi=\frac{m\omega}{2\hbar},$ (20)

in which $x$ and $y$ are the spatial coordinates of the relative motion. By
multiplying the Eq. (20) with $\gamma^{t}\otimes\gamma^{t}$ from left (note
that $\left(\gamma^{t}\otimes\gamma^{t}\right)^{2}$ gives $4\times 4$
dimensional unit matrix), one can obtain the following $4\times 4$ dimensional
matrix equation,

$\displaystyle\left(\begin{array}[]{cccc}\varepsilon-M&\partial_{-}&-\partial_{-}&0\\\
-\partial_{+}&\varepsilon&0&-\partial_{-}\\\
\partial_{+}&0&\varepsilon&\partial_{-}\\\
0&\partial_{+}&-\partial_{+}&\varepsilon+M\end{array}\right)\psi\left(\overrightarrow{r}\right)$
(25)
$\displaystyle\quad\quad\quad\quad\quad\quad+\left(\xi-\kappa\right)\left(\begin{array}[]{cccc}0&r_{-}&r_{-}&0\\\
r_{+}&0&0&r_{-}\\\ r_{+}&0&0&r_{-}\\\
0&r_{+}&r_{+}&0\end{array}\right)\psi\left(\overrightarrow{r}\right)=0,$ (30)
$\displaystyle M=\frac{2mc}{\hbar},\quad\partial_{\pm}=\partial_{x}\pm
i\partial_{y},\quad r_{\pm}=x\pm iy.$ (31)

In order to construct all possible spin eigen-states of the system that under
scrutiny we can transform the spacetime background into the polar spacetime
background with the help of the well-known expressions in follows
guvendi2019exact ; guvendi2020relativistic ,

$\displaystyle\partial_{\mp}=e^{\mp
i\phi}\left(\mp\frac{i}{r}\partial_{\phi}+\partial_{r}\right),\quad
r_{\mp}=re^{\mp i\phi},$

here, $\partial_{+}$ and $\partial_{-}$ stand for the spin raising and spin
lowering operators, respectively, and $r$ is the radial distance between the
particles. Only for the transformed bi-spinor,

$\displaystyle\leavevmode\resizebox{455.6629pt}{}{$\psi\left(\overrightarrow{r}\right)\Longrightarrow\left(\varphi_{1}\left(r\right)e^{i\left(s-1\right)\phi},\varphi_{2}\left(r\right)e^{is\phi},\varphi_{3}\left(r\right)e^{is\phi},\varphi_{4}\left(r\right)e^{i\left(s+1\right)\phi}\right)^{\textbf{T}}$},$

we can obtain a set of coupled equations for the components,

$\displaystyle\varepsilon\psi_{1}\left(r\right)-M\psi_{2}\left(r\right)+2\partial_{r}\psi_{3}\left(r\right)+2ur\psi_{4}\left(r\right)=0,$
$\displaystyle\varepsilon\psi_{2}\left(r\right)-M\psi_{1}\left(r\right)+\frac{2s}{r}\psi_{3}\left(r\right)=0,$
$\displaystyle\varepsilon\psi_{3}\left(r\right)+\frac{2s}{r}\psi_{2}\left(r\right)-2\left(\frac{1}{r}+\partial_{r}\right)\psi_{1}\left(r\right)=0,$
$\displaystyle\varepsilon\psi_{4}\left(r\right)-2ur\psi_{1}\left(r\right)=0,$
$\displaystyle\psi_{1}\left(r\right)=\varphi_{1}\left(r\right)+\varphi_{4}\left(r\right),\quad\psi_{2}\left(r\right)=\varphi_{1}\left(r\right)-\varphi_{4}\left(r\right),$
$\displaystyle\psi_{3}\left(r\right)=\varphi_{2}\left(r\right)-\varphi_{3}\left(r\right),\quad\psi_{4}\left(r\right)=\varphi_{2}\left(r\right)+\varphi_{3}\left(r\right)$
$\displaystyle u=\left(\xi-\kappa\right),$ (32)

here, $s$ represents to total spin of the composite system formed by a
fermion-antifermion pair interacting through Dirac oscillator interaction in
the presence of an external uniform magnetic field. Provided that the spins of
these fermions are anti-symmetric with respect to each other, we can introduce
a dimensionless independent variable that reads $z=ur^{2}$. In terms of the
variable $z$, the equation system in Eq. (32) becomes as follows,

$\displaystyle\leavevmode\resizebox{446.2658pt}{}{$\varepsilon\psi_{1}\left(z\right)-M\psi_{2}\left(z\right)+4u\sqrt{\frac{z}{u}}\partial_{z}\psi_{3}\left(z\right)-2u\sqrt{\frac{z}{u}}\psi_{4}\left(z\right)=0$},$
$\displaystyle\varepsilon\psi_{2}\left(z\right)-M\psi_{1}\left(z\right)=0,$
$\displaystyle\varepsilon\psi_{3}\left(z\right)-\frac{2}{\sqrt{\frac{z}{u}}}\psi_{1}\left(z\right)-4u\sqrt{\frac{z}{u}}\partial_{z}\psi_{1}\left(z\right)=0,$
$\displaystyle\varepsilon\psi_{4}\left(z\right)-2u\sqrt{\frac{z}{u}}\psi_{1}\left(z\right)=0.$
(33)

One can see that it is not difficult to obtain an exact solution of this set
of equations written in the most symmetric form. We will obtain that this
solution gives relativistic Landau levels of a spinless composite system
consisting of a fermion-antifermion pair interacting through Dirac oscillator
interaction, without requires any approximation.

Figure 1: The dependence of the total energy on the $\overline{\omega}$. Here,
$m=c=\hbar=1$ and $0\leq\left(\omega-\frac{\omega_{c}}{2}\right)\leq 1$.

##  4 Non-perturbative energy spectrum

Solving the equation system in Eq. (33) in favour of $\psi_{1}\left(z\right)$
one can acquire the following $2^{nd}$ order wave equation,

$\displaystyle\leavevmode\resizebox{460.35788pt}{}{$\partial^{2}_{z}\psi_{1}\left(z\right)+\frac{1}{z}\partial_{z}\psi_{1}\left(z\right)+\left(\frac{\varepsilon^{2}-M^{2}}{16uz}-\frac{1}{4z^{2}}-\frac{1}{4}\right)\psi_{1}\left(z\right)=0$}.$

This wave equation can be reduced to the well-known shape of the Whittaker
differential equation via the ansatz that reads
$\psi_{1}\left(z\right)=\frac{1}{\sqrt{z}}\varsigma\left(z\right)$,

$\displaystyle\partial^{2}_{z}\varsigma\left(z\right)+\left(\frac{\varepsilon^{2}-M^{2}}{16uz}-\frac{1}{4}\right)\varsigma\left(z\right)=0.$
(34)

Solution function of Eq. (34) is obtained as follows guvendi2020relativistic ;
book:874771 ; dernek2018relativistic ,

$\displaystyle\varsigma\left(z\right)=NW_{\varpi,\tau}(z),\quad\varpi=\frac{\varepsilon^{2}-M^{2}}{16uz},\quad\tau=\frac{1}{2},$

in which $N$ is the normalization constant. However, this solution function
can only be reduced to polynomial of degree $n\geq 0$ with respect to the
variable $z$ provided that the following condition holds
guvendi2020relativistic ; book:874771 ; dernek2018relativistic ,

$\displaystyle\frac{1}{2}+\tau-\varpi=-n,$ (35)

here, $n$ is the principal quantum number (non-negative integer). Now, one can
immediately obtain a non perturbative spectrum in energy domain with the help
of the expressions given in Eq. (34) and Eq. (35), since Eq. (35) leads to the
quantization condition for the formation of the composite system that under
scrutiny,

$\displaystyle E_{n}=\pm
2mc^{2}\sqrt{1+\frac{\overline{\omega}\hbar}{2mc^{2}}\left(n+1\right)},$
$\displaystyle\overline{\omega}=4\left(\omega-\frac{\omega_{c}}{2}\right),\quad\omega_{c}=\frac{|e|B_{0}}{mc},$
(36)

here, $\omega_{c}$ stands for the cyclotron frequency guvendi2020relativistic
; bermudez2007mesoscopic . The Eq. (36) gives the relativistic Landau levels
of a static and spinless composite system formed by a fermion-antifermion pair
interacting through Dirac oscillator interaction. This non-perturbative energy
spectrum shows that the system that under scrutiny behaves like a single
relativistic quantum oscillator carrying total rest mass of the composite
system. In Eq. (36), one can see that the second term in square root does not
vanish even for the ground state of such a system. We can also see that the
oscillation of this composite system will stop if and only if
$\omega=\omega_{c}/2$ holds for $e<0$ and the total energy of the system
becomes equal to the total rest mass energy of the particles when
$\omega=\omega_{c}/2$. Also, in terms of the variable $z$, we can easily
obtain the components of the bi-spinor as in the following,

$\displaystyle\psi_{1}\left(z\right)=N\frac{W_{\varpi,\tau}(z)}{\sqrt{z}},$
$\displaystyle\psi_{2}\left(z\right)=N\frac{M}{\varepsilon\sqrt{z}}W_{\varpi,\tau}(z),$
$\displaystyle\psi_{3}\left(z\right)=N\left[\frac{2}{\varepsilon\sqrt{\frac{1}{u}}}+\frac{4\sqrt{uz}}{\varepsilon}\left(\frac{1}{2}-\frac{\varpi}{z}\right)\right]W_{\varpi,\tau}(z)$
$\displaystyle-N\frac{4\sqrt{u}}{\varepsilon z}W_{\varpi+1,\tau}(z),$
$\displaystyle\psi_{4}\left(z\right)=-N\frac{2\sqrt{u}}{\varepsilon}W_{\varpi,\tau}(z),$
(37)

which satisfy the coupled equations in Eq. (33), of course.

##  5 Results and Discussion

In this paper, we introduce a unique model for a composite system formed by a
fermion-antifermion pair interacting through Dirac oscillator interaction in
the presence of an external uniform magnetic field. This model based on an
exact solution of the corresponding form of a fully-covariant two body Dirac
equation (one-time) including spin algebra spanned by Kronocker productions of
the Dirac matrices. The translational symmetry of the interested system allows
to study in $2+1$ dimensional flat Minkowski spacetime background. We
transform the spacetime background into the polar spacetime background so that
we can exploit the angular symmetry. This transformation provides to obtain an
exactly solvable $2^{nd}$ order wave equation for such a composite system (see
also moshinsky1993barut ; moshinsky1991anomalous ). We solve this wave
equation and arrive at a non-perturbative spectrum (in closed-form) in energy
domain. The obtained spectrum (Eq. (36)) shows that, the total energy
($E_{n}$) of the system closes to total rest mass energy ($2mc^{2}$) of this
composite system (static) when both of the external magnetic field and
oscillator frequency close to zero. It is clear that in Eq. (36), the term
associated with the both external magnetic field and oscillator frequencies
does not vanish for such a composite system in the ground state ($n=0$). We
can also see that, the obtained total energy value closes to the total rest
mass energy of the system when $\omega\approx\frac{\omega_{c}}{2}$ and the
oscillation of this composite system will stop if and only if
$\omega=\frac{\omega_{c}}{2}$ is satisfied. The obtained energy spectrum in
Eq. (36) shows that the system that under scrutiny behaves like a single
relativistic quantum oscillator carrying total rest mass of the composite
system in the presence of external uniform magnetic field. The dependence of
the total energy on the $\overline{\omega}$ can be seen in Fig. $1$. As a
result, our unique presentation in this manuscript provides to obtain the
relativistic Landau levels of a spinless composite system consisting of a
fermion-antifermion pair interacting through Dirac oscillator interaction. We
think that the obtained non-perturbative results in this paper can give info
about the quark-antiquark systems and thus mass formula for mesons (see also
moshinsky1993barut ).

###### Acknowledgements.

The author thanks Özge SAKARYA ÇINKI for style suggestions.

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