A note on the zeros of zeta and $L$-functions

TL;DR

This paper provides a new proof for the best known bounds on the argument of the Riemann zeta-function assuming the Riemann hypothesis and extends these bounds to a broad class of $L$-functions, also exploring related zero and vanishing properties.

Contribution

It introduces a simplified proof of the sharpest bounds for $S(t)$ and generalizes these bounds to various $L$-functions from automorphic representations.

Findings

Established the sharpest known bounds for $S(t)$ under RH

Extended bounds to a wide class of $L$-functions

Provided results on zero order and height of the lowest zero

Abstract

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound for a large class of $L$-functions including those which arise from cuspidal automorphic representations of GL($m$) over a number field. We also prove a number of related results including bounding the order of vanishing of an $L$-function at the central point and bounding the height of the lowest zero of an $L$-function.