An explicit upper bound of the argument of Dirichlet L-functions on the generalized Riemann hypothesis (Analytic Number Theory : Distribution and Approximation of Arithmetic Objects)

Abstract

We prove an explicit upper bound of the function S(t, $chi$), defined by the argument of Dirichlet L-functions attached to a primitive Dirichlet character $chi$(mod q>1). An explicit upper bound of the function S(t), defined by the argument of the Riemann zeta-function, have been obtained by A. Fujii [1]. Our result is obtained by applying the idea of Fujii s result on S(t). The constant part of the explicit upper bound of S(t, $chi$) in this paper does not depend on $chi$. Our proof does not covor the case q=1 and indeed gives a better bound than the one of Fujii that covers the case q=1.