On a new approach to the Riemann hypothesis

TL;DR

This paper explores a new approach to the Riemann hypothesis by analyzing the residues of certain series at hypothetical zeros off the critical line, deriving asymptotic relations that could impact the hypothesis.

Contribution

It establishes an asymptotic relation connecting residues of series at potential nontrivial zeros to a continuous function, offering a novel perspective on the Riemann hypothesis.

Findings

Derived asymptotic relation for residues at zeros off the critical line

Connected residues to a continuous function in rational parameter p

Implications discussed for the validity of the Riemann hypothesis

Abstract

Suppose that the Riemann hypothesis is false and $\rho_{*} = 1/2 + \eta_{*} + i \gamma_{*}$, $\eta_{*} > 0$, is a nontrivial zero of the Riemann $\zeta$-function off the critical line. Under the negation of the Riemann hypothesis for the Riemann $\zeta$-function, we establish an asymptotic relation (as $\gamma_{*} \to \infty$) which relates the residues of the series $\sum_{n \geq 1} \Lambda(n) e^{- 2\pi i p n } n^{-s}$ at $s =$ corresponding nontrivial zeros of some Dirichlet $L$-functions to some function, valid for any rational number $p = a/b < 1$ with $b \ll \log \gamma_{*}$. This related function is continuous in $p$ and we mention its implication to the Riemann hypothesis.