Summary
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.
Mathematics Subject Classifications (2010). 11Y60, 65B10
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Acknowledgements
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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Sondow, J. (2010). New Vacca-Type Rational Series for Euler’s Constant γ and Its “Alternating” Analog \(\ln \frac{4}{\pi }\) . In: Chudnovsky, D., Chudnovsky, G. (eds) Additive Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68361-4_23
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DOI: https://doi.org/10.1007/978-0-387-68361-4_23
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