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value of the Riemann zeta function at $s=2$

Major Section: 
Reference
Type of Math Object: 
Theorem

Mathematics Subject Classification

11M99 no label found42A16 no label found

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Riemann proved that $\zeta(s)$ is extensible analitically over $\mathbb{C}$ like a meromorphic function having a simple pole with residue $1$ at $s=1$, trivial zeros $\sigma\in2\mathbb{Z}^-$ and non-trivial zeros lying in the critical band.
In a recent work in PM owned by Mr. Alozano, he performed the calculation of $\zeta(2)$ using Fourier series and the well-known Parseval's identity. He got $\zeta(2)=\frac{\pi^2}{6}$, a trascendent irrational number. But not always neither we can evaluate $\zeta(s)$ by using the mentioned technic nor this one is a irrational number. That is the case about $\zeta(0)$. The more expeditive way to catch it was given by Mr. Hammick in its entry: "formulae for zeta in the critical strip". In the semi-plane $\Re(s)>-1$, we have the formula
$$\zeta(s)=\frac{1}{s-1}+\frac{1}{2}-s\int_1^\infty\frac{((x))}{x^{s+1}}dx$$
where $((x))=x-\left\lfloorx\right\rfloor-\frac{1}{2}=<x>-\frac{1}{2}$
Here $<\cdot>$ stands for the fractional function. Next, taking $s=0$,
we get trivially $\zeta(0)=-\frac{1}{2}$, a rational number and, maybe the unique one.
Pedro

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