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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)].

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