Wrong proof of the functional equation for $ \zeta (s) $ but why is the result correct?

Question

If I introduce the function $ f(x)= |x|^{s-1} $ inside Poisson summatory formula and use the fact that

$$ \sum_{n=-\infty}^{\infty}|n|^{s-1}=2\zeta (1-s) $$

If I combine this expression in the Poisson sum formula and the Mellin transform

$$ \int_{0}^{\infty}t^{s-1}\cos(at)dt = \Gamma (s)\cos(\pi s /2)/a^{s} $$

I manage to prove Riemann's functional equation

$$ \zeta (1-s)= 2(2\pi )^{-s}\Gamma(s)\cos(\pi s/2)\zeta (s) $$

But I have proved this using BAD mathematics then why is the result correct? How is that possible ?

Answer 1

For a good proof, and in particular for an easy one, see the article of Knopp and Robins here. They present a new, simple proof, based upon Poisson summation.