An explicit upper bound of the argument of Dirichlet $L$-functions on the generalized Riemann hypothesis

TL;DR

This paper establishes an explicit upper bound for the argument of Dirichlet L-functions, extending previous bounds for the Riemann zeta-function, under the generalized Riemann hypothesis, with a constant independent of the character.

Contribution

It provides a new explicit upper bound for the argument of Dirichlet L-functions that does not depend on the specific primitive character, extending Fujii's approach.

Findings

Explicit upper bound for $S(t, ext{chi})$ established

Constant part of the bound is independent of the character

Method extends Fujii's approach to Dirichlet L-functions

Abstract

We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument of Dirichlet $L$-functions. An explicit upper bound of the function $S_1(t)$, defined by the integral of the argument of the Riemann zeta-function, have already been obtained by A. Fujii. Our result is obtained by applying an idea of Fujii's result on $S_1(t)$. The constant part of the explicit upper bound of $S(t,\chi)$ in this paper does not depend on a primitive Dirichlet character $\chi\pmod {q>1}$.