In this blog entry we will derive the functional equation for the Riemann Zeta Function. It extends the Zeta function to the entire complex plane except for the point which is a simple pole. We have already found an analytic continuation for the Zeta function through the Euler Maclaurin summation formula . There, we were able to extend it´s domain to the left side of the complex plane step by step increasing the order of the Euler Maclaurin formula. The functional equation enables us to extend the Zeta function to the entire complex domain at once. We will start by first introducing the Poisson summation formula which is a key ingredient in the derivation. In the end of the post we will show one small branch of it´s applicability proving the result , which we have extensively used computing integrals. Poisson Summation Formula Let be a continuous function of define...
