On a functional equation

Abstract

In this article, the author derives a functional equation (1) η ( s ) = [ ( π 4 ) s - 1 2 2 π Γ ( 1 - s ) sin ( π s 2 ) ] η ( 1 - s ) of the analytic function η( s) which is defined by (2) η ( s ) = 1 - s - 3 - s - 5 - s + 7 - s + … for complex variable s with Re s > 1, and is defined by analytic continuation for other values of s. The author proves (1) by Ramanujan identity (see [1], [3]). Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.