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

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

2003-04-02

arxiv.org/abs/math/0304021v1

Ramanujan J. 12 (2006) 225-244

Mathematics

Number Theory

AmSTeX, 24 pages, submitted for publication

Scientific paper

10.1007/s11139-006-0075-1

The aim of the paper is to relate computational and arithmetic questions about Euler's constant $\gamma$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical formula for $\gamma$ due to Ramanujan, together with Vacca's and Gosper's series for $\gamma$, 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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