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A class of series acceleration formulae for Catalan’s constant. (English) Zbl 0946.11032

The author shows that the function \[ T(r)=\int^{r\pi}_0 \log( \tan \theta) d\theta, \quad 0\leq r\leq{1\over 2}, \] satisfies a reflection formula and a multiplication formula. These are used to find infinitely many series for Catalan’s constant and some other Dirichlet \(L\)-series, of which Ramanujan’s formula \[ G={\pi\over 8}\log (2+\sqrt 3)+ {3\over 8} \sum^\infty_{k=0} {1\over (2k+1)^2 {2k\choose k}} \] is the simplest. The paper concludes with a description of the ideas that motivated the present work. A list of linear relations amongst various values of \(T\) with rational arguments having denominator at most 20 is also given.

MSC:

11Y60 Evaluation of number-theoretic constants
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