🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

For $s>1$, $\zeta (s)=\frac{s}{s-1}-s{\int }_1^{\infty }\frac{\{x\}}{x^{s+1}}dx$. The integral converges for $s>0$, and ${lim}_{s\to 1^{+}}(s-1)\zeta (s)=1$ (Zeta-Function has a simple pole at s=1 with residue 1).