Summation of Series Defined by Counting Blocks of Digits

Details Summation of Series Defined by Counting Blocks of Digits Summation of Series Defined by Counting Blocks of Digits

2005-12-16

arxiv.org/abs/math/0512399v2

Journal of Number Theory 123 (2007) 133-143

Mathematics

Number Theory

12 pages, Introduction expanded, references added, accepted by J. Number Theory

Scientific paper

10.1016/j.jnt.2006.06.001

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 $\sum_{n \geq 1}
s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2$. We recover this previous
result of Sondow in math.NT/0508042 and provide several generalizations.

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