Summation of Series Defined by Counting Blocks of Digits

Jean-Paul Allouche; Jeffrey Shallit; Jonathan Sondow

doi:10.1016/j.jnt.2006.06.001

paper

Summation of Series Defined by Counting Blocks of Digits

arXiv:math/0512399 · doi:10.1016/j.jnt.2006.06.001

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 $\sum_{n \geq 1} s_{2} (n) / (2 n (2 n + 1)) = (γ + lo g \frac{4}{π}) /2$ . We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.

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