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arXiv · 2205.04576

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

William D. Banks
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Abstract

For any real $β_0\in[\tfrac12,1)$, let ${\rm GRH}[β_0]$ be the assertion that for every Dirichlet character $χ$ and all zeros $ρ=β+iγ$ of $L(s,χ)$, one has $β\leβ_0$ (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 $ζ(s)$. No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.

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William D. Banks. 2023-01-31. Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function. https://arxiv.org/abs/2205.04576

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