A Faster Product for Pi and a New Integral for ln(Pi/2)

Abstract

From a global series for the alternating zeta function, we derive an infinite product for pi that resembles the product for $e^{\gamma }$ ($\gamma$ is Euler's constant) in math.CA/0306008. (An alternate derivation accelerates Wallis's product by Euler's transformation.) We account for the resemblance via an analytic continuation of the polylogarithm. An application is a 1-dim. analog for ln(pi/2) of the 2-dim. integrals for ln(4/pi) and $\gamma$ in math.CA/0211148.

7 pages, 1 figure, revision accepted for publication by Amer. Math. Monthly contains a product for e due to J. Guillera and two additional references, one by Hasse