On values of the Riemann zeta function at positive integers

Abstract

We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2n) = ηnπ2n, we obtain the generating functions of the sequences ηn and (−1)nηn. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2n) and ζ(2n + 1). Furthermore, we prove some series equations for $$ {\sum}_{k=1}^{\infty }{\left(-1\right)}^{k-1}\upzeta \left(p+k\right)/k. $$