# Math Help - Riemann Zeta Function

1. ## Riemann Zeta Function

Show that $\zeta(s) < 0$ for $0 < s < 1$.

2. The Laurent expansion of the $\zeta(*)$ around $s=1$ is...

$\zeta(s)= \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!}\cdot \gamma_{n}\cdot (s-1)^{n}$ (1)

... where the $\gamma_{n}$ are the so called 'Stieltjes constants'. For $s$ 'not too far' from 1 the term $\frac{1}{s-1}$ is dominant, so that $\zeta(*)$ is negative in $0 < s < 1$. The nearest zero on the real axis is at $s=-2$...

Kind regards

$\chi$ $\sigma$

P.S. That's a graphich I've made about a year ago using the (1)...