Asymptotic properties of zeros of Riemann zeta function

TL;DR

This paper investigates the asymptotic behavior of the zeros of the Riemann zeta function, aiming to understand their intrinsic properties and whether other sequences share similar asymptotic relations.

Contribution

It introduces a new asymptotic relation for the zeros of the zeta function and explores the uniqueness of such sequences with this property.

Findings

Derived an asymptotic relation involving the zeros of the zeta function.

Identified specific coefficients related to Bernoulli and Euler numbers.

Raised questions about the uniqueness of sequences satisfying the asymptotic property.

Abstract

We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let $(z_k)_{k\in \mathbb{N}}$ be the sequence of nontrivial zeros of $\zeta(s)$ with positive imaginary part. We write $z_k= 1/2+i\tau_k$ (RH says that these $\tau_k$ are all real). Then the sequence $(\tau_k)_{k\in \mathbb{N}},$ satisfies the following asymptotic relation \[\sum_{k\in\mathbb{N}}\frac{2x}{x^2+\tau_k^2}\simeq \frac12\log\frac{x}{2\pi}+\sum_{n=1}^\infty \frac{a_n}{x^n},\,\,x\to +\infty\] where $a_{2n+1}=2^{-2n-2}(8-E_{2n})$, $a_{2n}=(1-2^{-2n+1})B_{2n}/(4n).$ Are there other sequences $(\alpha_k)_{k\in \mathbb{N}},$ of real or complex numbers enjoying this property? These problems are addressed in this note.