Series over Bessel functions as series in terms of Riemann's zeta function

TL;DR

This paper derives new formulas connecting series of sine and cosine functions to the Hurwitz and Riemann zeta functions, providing finite sum representations using advanced summation techniques.

Contribution

It introduces novel methods to express series over Bessel functions as series involving the Riemann zeta function, expanding the analytical tools available for such series.

Findings

Derived sums of sine and cosine series via Hurwitz zeta function

Established finite sum formulas for series involving Riemann zeta function

Utilized summation formulas to connect trigonometric series with zeta functions

Abstract

Relying on the Hurwitz formula, we find sums of the series over sine and cosine functions through the Hurwitz zeta function. Using another summation formula for these trigonometric series, we find finite sums of some series over the Riemann zeta function.