Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 150 / 2 / b / Solution

The functional equation isEquivalently,

For a proof, letThe Poisson summation formula applied to a Gaussian function gives the theta transformationThe standard Gamma function integral and termwise integration initially give, for $\Re s>1$,Split the integral at one, substitute $u\mapsto 1/u$ in the lower half, and use the theta transformation. The result isThe integral is an entire function of $s$ because $\Theta (u)-1$ decays exponentially. The right side is visibly invariant under $s\mapsto 1-s$, proving both the analytic continuation and the Functional equation of the Riemann zeta function. This is the Mellin representation of the completed Riemann zeta function.