Some identities for the Riemann zeta-function

Abstract

Several identities for the Riemann zeta-function $ζ(s)$ are proved. For example, if $s = σ+ it$ and $σ> 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})ζ(s)\over s}|^2dt = {π\overσ}(1 - 2^{1-2σ})ζ(2σ). $$