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 $\Re(s)>0$ minus $s=1$.

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

We show that it is analytic for $\Re(s)>0$.

For $n \ge 1$, let:

Here $\ll$ is the order notation.

By the Mean Value Theorem, for some $n \le \theta \le n+1$:


 * $\left({n + 1}\right)^s - n^s = s \theta^{s-1} \le s \left({n + 1}\right)^{s-1}$

Thus if $s = \sigma + i t$:


 * $\left\vert{a_n}\right\vert \le \left\vert{\dfrac s {n^{s+1} } } \right\vert = \dfrac {\sigma^2 + t^2} {n^{\sigma + 1} }$

Since:


 * $\displaystyle \zeta \left({s}\right) = \frac s {s - 1} - \sum_{n \mathop \ge 1} a_n$

it follows that this representation converges absolutely uniformly on $\Re \left({s}\right) > 0$.

Thus by Uniform Limit of Analytic Functions is Analytic $\zeta \left({s}\right)$ is analytic for $\Re \left({s}\right) > 0$ and $s \ne 1$.

Also see

 * Analytic Continuations of Riemann Zeta Function to Right Half-Plane