Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted L-Functions
Abstract
We derive explicit formulas for finite sums of powers of cotangents, both at rational and arbitrary arguments, together with their alternating counterparts, weighted by additive characters. Our method relies on Apostol--Bernoulli polynomials and on the partial fraction expansion of trigonometric functions. As applications, we solve the problem posed by Berndt, Kim and Zaharescu \cite{BKZ:2023} on evaluating ∑csc2m and ∑sec2m at rational arguments, recover as a special case the classical Berndt--Yeap formula, and relate our identities to spectral zeta functions of circulant graphs. Furthermore, to systematically evaluate higher-order derivatives arising from the Lipschitz summation formula, we introduce a novel sequence of polynomial coefficients cm,j(β), termed \emph{deformed cotangent coefficients}, and establish their recursive and algebraic properties. Using these coefficients, we establish a general additive-and-multiplicatively twisted character mean value theorem for Lerch zeta functions, significantly extending the framework of Bayad and Raouj to incorporate additive character twists.
Related
- A Quantum-Inspired Dequantization Method for Diagonally Weighted Matrix Functions: Application to Learning with Optimized Random Features
- Distribution of central values of Hecke L-functions of n-th order characters
- Rook characters as symmetric functions
- Irrationality of values of L-functions for real Dirichlet characters
- Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields
BibTeX
@misc{bayad2026finite,
title = {{Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted $L$-Functions}},
author = {Abdelmejid Bayad and Salah Eddine Rihane},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01801},
eprint = {2611.01801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01801},
}
RIS
TY - UNPB
TI - Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted L-Functions
AU - Abdelmejid Bayad
AU - Salah Eddine Rihane
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01801
UR - https://arxiv.org/abs/2611.01801
AB - We derive explicit formulas for finite sums of powers of cotangents, both at rational and arbitrary arguments, together with their alternating counterparts, weighted by additive characters. Our method relies on Apostol--Bernoulli polynomials and on the partial fraction expansion of trigonometric functions. As applications, we solve the problem posed by Berndt, Kim and Zaharescu \cite{BKZ:2023} on evaluating ∑csc^2m and ∑sec^2m at rational arguments, recover as a special case the classical Berndt--Yeap formula, and relate our identities to spectral zeta functions of circulant graphs. Furthermore, to systematically evaluate higher-order derivatives arising from the Lipschitz summation formula, we introduce a novel sequence of polynomial coefficients c_m,j(β), termed \emph{deformed cotangent coefficients}, and establish their recursive and algebraic properties. Using these coefficients, we establish a general additive-and-multiplicatively twisted character mean value theorem for Lerch zeta functions, significantly extending the framework of Bayad and Raouj to incorporate additive character twists.
ER -