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Summation of Series Defined by Counting Blocks of Digits

Jean-Paul Allouche·Jeffrey Shallit·Jonathan Sondow
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Abstract

We discuss the summation of certain series defined by counting blocks of digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (γ+ \log \frac{4}π)/2$. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.

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BibTeXRIS

Jean-Paul Allouche, Jeffrey Shallit, Jonathan Sondow. 2006-06-02. Summation of Series Defined by Counting Blocks of Digits. https://doi.org/10.1016/j.jnt.2006.06.001

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