On some series connected with Riemann zeta function

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 $$ \sum_{n\in\mathbb{Z}}(M/N)^{(J+1)(n+w)} \varphi_{J,M,N}((M/N)^{n+w})=J!.$$