LITHUANIAN MATHEMATICAL JOURNAL, vol.60, no.1, pp.9-24, 2020 (SCI-Expanded)
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 zeta(2n) = eta(n)pi(2n), we obtain the generating functions of the sequences eta(n) and (-1)(n)eta(n). Using the Riemann-Lebesgue lemma, we give recurrence relations for zeta(2n) and zeta(2n + 1). Furthermore, we prove some series equations for n-ary sumation Sigma(infinity)(k=1) 1(-1)(k-1)zeta(p + k)/k.