Jean-Paul Allouche, Jeffrey Shallit, Jonathan Sondow
Published 2005-12-16, updated 2006-06-02Version 2
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.
Comments: 12 pages, Introduction expanded, references added, accepted by J. Number Theory
Journal: Journal of Number Theory 123 (2007) 133-143
On the $b$-ary expansions of $\log (1 + \frac{1}{a})$ and ${\mathrm e}$
Infinite products with strongly $B$-multiplicative exponents
On the expansions of real numbers in two integer bases
![QR code for arXiv:math/0512399 [math.NT]](http://oss.arxitics.com/images/favicon.svg)