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

Bernd C. Kellner

Published 2012-09-19, updated 2014-10-28Version 2

We show for even positive integers $n$ that the quotient of the Riemann zeta values $\zeta(n+1)$ and $\zeta(n)$ satisfies the equation $$\frac{\zeta(n+1)}{\zeta(n)} = (1-\frac{1}{n}) (1-\frac{1}{2^{n+1}-1}) \frac{\mathcal{L}^\star(\mathfrak{p}_n)}{\mathfrak{p}_n'(0)},$$ where $\mathfrak{p}_n \in \mathbb{Z}[x]$ is a certain monic polynomial of degree $n$ and $\mathcal{L}^\star: \mathbb{C}[x] \to \mathbb{C}$ is a linear functional, which is connected with a special $L$-function. There exists the decomposition $\mathfrak{p}_n(x) = x(x+1) \mathfrak{q}_n(x)$. If $n = p+1$ where $p$ is an odd prime, then $\mathfrak{q}_n$ is an Eisenstein polynomial and therefore irreducible over $\mathbb{Z}[x]$.