Some series involving the Euler zeta function

Abstract

In this paper, using the Boole summation formula, we obtain a new integral representation of $n$-th quasi-periodic Euler functions $\overline{E}_n(x)$ for $n=1,2,\ldots.$ We also prove several series involving Euler zeta functions $\zeta_{E}(s),$ which are analogues of the corresponding results by Apostol on some series involving the Riemann zeta function $\zeta(s).$