On quotients of Riemann zeta values at odd and even integer arguments

Abstract

We show for even positive integers n that the quotient of the Riemann zeta values zeta(n + 1) and zeta(n) satisfies the equation zeta(n + 1)/zeta(n) = (1 - 1/n) (1 - 1/2(n+1) - 1) L-star(p(n))/p(n)'(0), where p(n) is an element of Z[x] is a certain monic polynomial of degree n and L-star : C[x] -> C is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition p(n)(x) = x(x + 1)q(n)(x). If n = p + 1 where p is all odd prime, then q(n) is an Eisenstein polynomial and therefore irreducible over Z[x] (C) 2013 Elsevier Inc. All rights reserved.