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).
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