Mathematics – Number Theory
Scientific paper
2005-06-16
Ramanujan J. 16 (2008) 247-270
Mathematics
Number Theory
21 pages, to appear in The Ramanujan Journal. Added Corollary 3.3, Lemma 3.1, Equations (5) and (33), all or part of Examples
Scientific paper
10.1007/s11139-007-9102-0
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $\pi$ and $e^\gamma.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.
Guillera Jesus
Sondow Jonathan
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