A new result under the negation of the Riemann hypothesis
Hisanobu Shinya
TL;DR
The paper investigates consequences of a false Riemann hypothesis by introducing the Mangoldt-based function $M(s,p)$ and expressing $M(s,a/b)$ in terms of Dirichlet $L$-functions. It develops a framework in which a would-be off-line nontrivial zero affects the poles of $M(s,p)$ across rational $p$, analyzed through contour integrals and residues. The main results consist of a key identity (Theorem main1) linking $M(s+\delta+\kappa,p)$ to $\zeta'/\zeta$ and Dirichlet data, and a second theorem (Theorem claim) detailing how potential poles of $M$ induce residue contributions and a continuity property in $p$. This approach aims to provide a workable pathway to obtain new insights about the zeros of $\zeta$ and related $L$-functions, potentially informing the falsity scenario for the Riemann hypothesis and guiding further research on zero distributions and Ramanujan sums.
Abstract
Suppose that the Riemann hypothesis for the Riemann $ζ$-function is false. There have yet been no workable results that the falsity of the Riemann hypothesis implies. The result of this article may be considered as more workable for the sake of deducing other new results on the Riemann hypothesis.