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:

$\zeta (0)$	$=\underset{s\to 0^{+}}{lim}\zeta (s)$
	$=\underset{s\to 0^{+}}{lim}{2}^s{\pi }^{s-1}\sin {\displaystyle \frac{\pi s}{2}}\mathrm{\Gamma }(1-s)\zeta (1-s)$
	$=\underset{s\to 0^{+}}{lim}{2}^s{\pi }^{s-1}\left({\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi s}{2}}\right)}^{2n+1}\right)\mathrm{\Gamma }(1-s)\left({\displaystyle \frac{1}{(1-s)-1}}+1-(1-s){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{(1-s)+1}}}𝑑x\right)$
	$=\underset{s\to 0^{+}}{lim}{2}^s{\pi }^{s-1}\left({\displaystyle \frac{\pi s}{2}}\right)\left({\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi s}{2}}\right)}^{2n}\right)\mathrm{\Gamma }(1-s)\left({\displaystyle \frac{1}{-s}}+1-(1-s){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{2-s}}}𝑑x\right)$
	$=\underset{s\to 0^{+}}{lim}{2}^s{\pi }^{s-1}\left({\displaystyle \frac{\pi }{2}}\right)\left(1+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi s}{2}}\right)}^{2n}\right)\mathrm{\Gamma }(1-s)s\left({\displaystyle \frac{-1}{s}}+1-(1-s){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{2-s}}}𝑑x\right)$
	$=\underset{s\to 0^{+}}{lim}{2}^{s-1}{\pi }^s\left(1+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi s}{2}}\right)}^{2n}\right)\mathrm{\Gamma }(1-s)\left(-1+s-s(1-s){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{2-s}}}𝑑x\right)$
	$=\left(\underset{s\to 0^{+}}{lim}{2}^{s-1}{\pi }^s\mathrm{\Gamma }(1-s)\left(-1+s-s(1-s){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{2-s}}}𝑑x\right)\right)\left(\underset{s\to 0^{+}}{lim}1+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi s}{2}}\right)}^{2n}\right)$
	$=\left(2^{0-1}{\pi }^0\mathrm{\Gamma }(1-0)\left(-1+0-0(1-0){\displaystyle {\int }_1^{\mathrm{\infty }}}{\displaystyle \frac{x-[x]}{x^{2-0}}}𝑑x\right)\right)\left(1+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{(2n+1)!}}{\left({\displaystyle \frac{\pi \cdot 0}{2}}\right)}^{2n}\right)$
	$=\left({\displaystyle \frac{1}{2}}\cdot 1\cdot \mathrm{\Gamma }(1)\cdot (-1+0-0)\right)\left(1+{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}0\right)$
	$={\displaystyle \frac{-1}{2}}$.

∎

Title	value of the Riemann zeta function at $s=0$
Canonical name	ValueOfTheRiemannZetaFunctionAtS0
Date of creation	2013-03-22 16:07:17
Last modified on	2013-03-22 16:07:17
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	20
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 11M06
Related topic	CriticalStrip
Related topic	FormulaeForZetaInTheCriticalStrip