Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

Journal of the Ramanujan Mathematical Society

Volume 35, Issue 4, December 2020  pp. 327–340.

Higher dimensional Dedekind sums and twisted mean values of Dirichlet L-series

Authors:  Mithun Kumar Das and Abhishek Juyal
Author institution:Harish-Chandra Research Institute (HBNI), Chhatnag Road, Jhunsi, Allahabad 211 019, Uttar Pradesh, India

Summary:  We provide an identity for evaluating products of trigonometric functions of the form sec{m}{2x} cot{2n}{x}, where m, n are positive integers. Using this identity, we are able to give a partial answer of a question raised by A. Straub (Ramanujan J. no. 41 [2016], 269--285). As an application of this identity, we evaluated formulas for special values of Don Zagier's higher dimensional Dedekind sums. We also study the mean values {2}/{φ (q)} ∑{{χ (mod q)}{χ (-1)=(-1){m}}} χ (c)L(m,χ )L(n,{χ}), where χ is a Dirichlet character modulo an odd integer q, c a positive integer coprime to q, φ(·) is the Euler's φ-function and m, n are positive integers. Moreover, we express the mean values in terms of higher dimensional Dedekind sums. For odd q and c = 1,2,4, we determine the mean values explicitly for some integers m, n, which generalize results of H. Liu (J. Number Theory 147 [2015], 172--183).

Contents   Full-Text PDF