On some series connected with Riemann zeta function

TL;DR

This paper introduces new classes of series related to the Riemann zeta-function, including integral representations and series involving theta functions and exponentials, with potential replacements for the zeta-function.

Contribution

It proposes two novel classes of evaluated series connected to the Riemann zeta-function and introduces functions that can substitute the zeta-function in similar series constructions.

Findings

Derived series involving theta functions and exponentials.

Established integral representations as generalized averages.

Provided explicit formulas and conditions for series evaluation.

Abstract

Using properties of the Riemann zeta-function we propose two new large classes of evaluated series. Incidentally the first class represents integrals as generalized average on very nonuniform sequences. The second class contains inter alia a lot of new series with the Jacoby theta-functions and rationals of the exponential function. Moreover we propose many functions that can replace the Riemann zeta-function in similar constructions. Two examples: 1) if $f(x)$ has period 1 and is in some Lipschitz class, we have for any natural $M>1$ $$ \ln M\cdot\int_0^1f(x)dx = \sum_{n\geq 1} \sum_{k=1}^{M-1}\left[\frac{1}{Mn-k}f\left(\frac{\ln (Mn-k)}{\ln M}\right)-\frac{1}{Mn} f\left(\frac{\ln (Mn)}{\ln M}\right)\right],$$ 2) if $\varphi_{J,M,N}(w) = (-1)^J\left(\frac{d^J}{dw^J}\right)\left(\frac{N}{e^{Nw}-1}-\frac{M}{(e^{Mw}-1}\right),$ where $J,M,N$ are integer, $M>N>1,$ $J\geq 0$ and for all $n\in\mathbb{Z}$, $\left(e^{M\left(M/N\right)^{n+w}} - 1\right) \left(e^{N\left(M/N\right)^{n+w}} -1 \right) \neq 0,$ we have $$ \sum_{n\in\mathbb{Z}}(M/N)^{(J+1)(n+w)} \varphi_{J,M,N}((M/N)^{n+w})=J!.$$