We find the sum of series of the form \sum\limits_{i = 1}^\infty {\frac{{f(i)}}{{{i^r}}}} or some special functions f. The above series is a generalization of the Riemann zeta function. In particular, we take f as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of π., Meher Jaban, Sinha Sneh Bala., and Obsahuje seznam literatury