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The Ramanujan Journal

, Volume 12, Issue 2, pp 225–244 | Cite as

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

  • Jonathan Sondow
  • Wadim Zudilin
Article

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.

Keywords

Euler’s constant q-logarithm Hypergeometric series Euler’s transform Irrationality 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.New YorkUnited States
  2. 2.Department of Mechanics and MathematicsMoscow Lomonosov State UniversityMoscowRussia

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