Easy proofs of Riemann's functional equation for \(\zeta(s)\) and of Lipschitz summation (Q2718950)

In the present paper the authors give a simple proof (using only Poisson summation) of the Lipschitz summation formula NEWLINE\[NEWLINE\sum_{n=1}^\infty(n-\alpha)^{s-1}e^{2\pi i\tau(n-\alpha)}={\Gamma(s) \over (-2\pi i)^s}\sum_{m=-\infty}^\infty {e^{2\pi i\alpha m}\over (\tau+m)^s},NEWLINE\]NEWLINE where \(0\leq\alpha<1\) and \(\tau\) lies in the upper half-plane. As an easy consequence they derive the functional equation for the Hurwitz zeta-function, given by \(\zeta(s,a)=\sum_{n=0}^\infty(n+a)^{-s}\) for \(\text{Re} s>1\), and, in particular, the Riemann zeta-function \(\zeta(s)=\zeta(s,1)\).