Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function

TL;DR

This paper demonstrates that the truth of a specific generalized Riemann hypothesis variant, ${ m GRH}[rac{9}{10}]$, can be determined solely by the distribution of zeros of the Riemann zeta function, without assumptions on other L-functions.

Contribution

It shows that the validity of ${ m GRH}[rac{9}{10}]$ depends only on the zeros of the Riemann zeta function, not on zeros of other Dirichlet L-functions.

Findings

${ m GRH}[rac{9}{10}]$ depends only on zeta zeros.

No conditions on nonprincipal Dirichlet L-function zeros.

Distributional properties of zeta zeros determine ${ m GRH}[rac{9}{10}]$.

Abstract

For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the assertion that for every Dirichlet character $\chi$ and all zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular, ${\rm GRH}[\frac12]$ is the Generalized Riemann Hypothesis). In this paper, we show that the validity of ${\rm GRH}[\frac{9}{10}]$ depends only on certain distributional properties of the zeros of the Riemann zeta function $\zeta(s)$. No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.