March 2023 On the mean square average of Dirichlet $L$-function over characters of odd parityin a special case
Neha Elizabeth Thomas, Arya Chandran, K. Vishnu Namboothiri
Funct. Approx. Comment. Math. 68(1): 41-57 (March 2023). DOI: 10.7169/facm/2020

Abstract

Evaluating the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of the same parity is an active problem in number theory. Here we explicitly evaluate $\sum_{\chi\text{ odd}}L(3,\chi)$ using certain trigonometric sums and Bernoulli polynomials and express the sum in terms of the Euler totient function $\phi$ and the Jordan totient function $J_s$.

Citation

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Neha Elizabeth Thomas. Arya Chandran. K. Vishnu Namboothiri. "On the mean square average of Dirichlet $L$-function over characters of odd parityin a special case." Funct. Approx. Comment. Math. 68 (1) 41 - 57, March 2023. https://doi.org/10.7169/facm/2020

Information

Published: March 2023
First available in Project Euclid: 18 November 2022

MathSciNet: MR4564863
Digital Object Identifier: 10.7169/facm/2020

Subjects:
Primary: 11M06
Secondary: 11L03 , 11L05

Keywords: $L$-functions , Bernoulli numbers , Euler totient function , Gauss sum , Jordan totient function , mean square averages , Ramanujan sum , Trigonometric sums

Rights: Copyright © 2023 Adam Mickiewicz University

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Vol.68 • No. 1 • March 2023
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