Kempner-like harmonic series

Abstract

Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to \infty}(\sum_{s_2(n) = k} 1/n) = 2 \log 2$, where $s_2(n)$ is the sum of the binary digits of the integer $n$. Then we propose two generalizations. One generalization addresses the case where $s_2(n)$ is replaced with $s_b(n)$, the sum of $b$-ary digits in base $b$: we prove that $\lim_{k \to \infty}\sum_{s_b(n) = k} 1/n = (2 \log b)/(b-1)$. The second generalization replaces the sum of digits in base $2$ with any block-counting function in base $2$, e.g., the function $a(n)$ of -- possibly overlapping -- $11$'s in the base-$2$ expansion of $n$, for which we obtain $\lim_{k \to \infty}\sum_{a(n) = k} 1/n = 4 \log 2$.