Asymptotic properties of zeros of Riemann zeta function

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 \[\sum_{k\in\mathbb{N}}\frac{2x}{x^2+τ_k^2}\simeq \frac12\log\frac{x}{2π}+\sum_{n=1}^\infty \frac{a_n}{x^n},\,\,x\to +\infty\] 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.

22 pages, 1 figure