A very short proof of the functional equation for ζ

This note contains a short proof of Riemann’s [R] functional equation for the zeta function. The argument is computationally similar to the Hankel contour argument that was one of Riemann’s own, and that is called the “second method” in Titchmarsh [T]. But it is made simpler to follow by the symmetries of sine: the fact that sine is a periodic and an odd function. In this form the argument seems to be new or at least not widely known. We begin with a simple lemma which uses integration by parts to introduce the sine (or hyperbolic sine). Lemma 1. For s > 1