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

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 $\chi (\mathrm{mod}q>1)$.