Integral Representation of Riemann Zeta Function in terms of Fractional Part

Theorem
Let $\zeta$ be the Riemann zeta function.

Let $s \in \C$ be a complex number with real part $\sigma > 1$.

Then
 * $\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$

where $\fractpart x$ denotes the fractional part of $x$.

Proof
We have:

Also see

 * Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part