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Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper

Jonathan Sondow·Wadim Zudilin
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Abstract

The aim of the paper is to relate computational and arithmetic questions about Euler's constant $γ$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical formula for $γ$ due to Ramanujan, together with Vacca's and Gosper's series for $γ$, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler's constant. The main tools are Euler-type integrals and hypergeometric series.

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Jonathan Sondow, Wadim Zudilin. 2003-04-02. Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper. https://doi.org/10.1007/s11139-006-0075-1

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