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 ${\sum }^{\prime }1/n$ 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 ${\lim }_{k\to \infty }{\sum }_{a(n)=k}1/n=4\log 2$.

A slightly shorter version has been accepted by the American Mathematical Monthly