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

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

The integral:


 * $\displaystyle \frac s {s-1} - s \int_1^\infty \left\{{x}\right\} x^{-s - 1} \ \mathrm d x$

defines an analytic continuation of $\zeta$ to the half-plane $\Re(s)>0$.

Proof
By Integral Representation of Riemann Zeta Function in terms of Fractional Part, it coincides with $\zeta(s)$ for $\Re(s)>1$.

Interchanging integral and derivative, one shows that the integral is analytic for $\Re(s)>0$.

Also see

 * Analytic Continuations of Riemann Zeta Function