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A faster product for \(\pi\) and a new integral for \(\ln \frac{\pi}{2}\). (English) Zbl 1159.11328
Summary: 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 [“An infinite product for \(e^\gamma\) via hypergeometric formulas for Euler’s constant, \(\gamma\)”, preprint,
url{arxiv:math/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 [Am. Math. Mon. 112, No. 1, 61–65 (2005; Zbl 1138.11356)].

11Y60 Evaluation of number-theoretic constants
11M35 Hurwitz and Lerch zeta functions
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