New Vacca-Type Rational Series for Euler’s Constant γ and Its “Alternating” Analog \(\ln \frac{4}{\pi }\)

  • Jonathan SondowEmail author


We recall a pair of logarithmic series that reveals ln(4 ∕ π) to be an “alternating” analog of Euler’s constant γ. Using the binary expansion of an integer, we derive linear, quadratic, and cubic analogs for ln(4 ∕ π) of Vacca’s rational series for γ. Using a generalization of Vacca’s series to integer bases b ≥ 2, due in part to Ramanujan, we extend Addison’s cubic, rational, base 2 series for γ to faster base b series. Open problems on further extensions of the results are discussed, and a history of the formulas is given.


Alternating Euler constant Acceleration of series Binary expansion Euler’s constant Generalized Euler constant Rational series Vacca’s series 



I am grateful to Stefan Krämer and Wadim Zudilin for valuable comments, and to Tanguy Rivoal for sending me a draft of [15].


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.New YorkUSA

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