Note on the mean absolute value theorem for the Dirichlet's $L$-function in the critical stripe

L. G. Arkhipova, V. N. Chubarikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics (Moscow)

Abstract: In the paper we are continued investigations on a generalization and a improvement of the R. T. Turganaliev's result by the deduction of the asymptotic formula for the mean-value of the Rieman's zeta-function in the critical stripe with the rest term, having the power in the reduction. We are found the asymptotics of Dirichlet's $L$-function in the critical stripe, which improves the R. T. Turganaliev's theorem on the zeta-function for all values of the real part ($1/2<\mathrm{Re}\, s\leq 1$). This result are got for the account of the different using of estimations of trigonometric sums on the base of the second derivative in the exponent.

Keywords: Dirichlet's characters, Dirichlet's functions, the zeta-sum twisted together with the Dirichlet's character.

UDC: 512.541

Received: 03.12.2020
Accepted: 21.02.2021

DOI: 10.22405/2226-8383-2018-22-1-67-75