value of the Riemann zeta function at $s=0$

Theorem.

Let $\zeta$ denote the meromorphic extension of the Riemann zeta function to the complex plane. Then $\zeta \mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}{\displaystyle \frac{\mathrm{-}\mathrm{1}}{\mathrm{2}}}$.

Proof.

Recall that one of the for the Riemann zeta function in the critical strip is given by

$$\zeta (s)=\frac{1}{s-1}+1-s{\int }_1^{\mathrm{\infty }}\frac{x-[x]}{x^{s+1}}𝑑x,$$

where $[x]$ denotes the integer part of $x$.

Also recall the functional equation

$$\zeta (s)=2^s{\pi }^{s-1}\sin \frac{\pi s}{2}\mathrm{\Gamma }(1-s)\zeta (1-s),$$

where $\mathrm{\Gamma }$ denotes the gamma function.

The only pole (http://planetmath.org/Pole) of $\zeta$ occurs at $s=1$. Therefore, $\zeta$ is analytic, and thus continuous, at $s=0$.

Let $\underset{s\to 0^{+}}{lim}$ denote the limit as $s$ approaches $0$ along any path contained in the region $\mathrm{Re}(s)>0$. Thus:

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