On the argument of $L$-functions

TL;DR

This paper establishes explicit bounds for certain $L$-function-related functions assuming the generalized Riemann hypothesis, providing new proofs for existing bounds on the argument of $L$-functions in terms of their analytic conductor.

Contribution

It offers an explicit bound for $S_1(t, ho)$ in terms of the analytic conductor, leading to an alternative proof of bounds for $S(t, ho)$ under GRH.

Findings

Derived explicit bounds for $S_1(t, ho)$ based on the analytic conductor.

Provided an alternative proof for bounds on $S(t, ho)$ assuming GRH.

Enhanced understanding of the argument of $L$-functions in analytic number theory.

Abstract

For $L(\cdot,\pi)$ in a large class of $L$-functions, assuming the generalized Riemann hypothesis, we show an explicit bound for the function $S_1(t,\pi)=\frac{1}{\pi}\int_{1/2}^\infty\log|L(\sigma+it,\pi)|\,d\sigma$, expressed in terms of its analytic conductor. This enables us to give an alternative proof of the most recent (conditional) bound for $S(t,\pi)=\frac{1}{\pi} \,arg\,L(\tfrac12+it,\pi)$, which is the derivative of $S_1(\cdot,\pi)$ at $t$.