On values of the Riemann zeta function at positive integers

DİL A., Boyadzhiev K. N., ALİYEV İ.

LITHUANIAN MATHEMATICAL JOURNAL, vol.60, no.1, pp.9-24, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 60 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.1007/s10986-019-09456-7
  • Journal Name: LITHUANIAN MATHEMATICAL JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, zbMATH, DIALNET
  • Page Numbers: pp.9-24
  • Keywords: Riemann zeta function, Apery's constant, Bernoulli numbers, generating function, polylogarithm, INTEGRALS
  • Akdeniz University Affiliated: Yes

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 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.