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