arXiv subjects

Claude Morin

Publications and source records attributed to Claude Morin.

2 recordsLinked to original sources

arXiv2024

Ellipsephic harmonic series revisited

Ellipsephic or Kempner-like harmonic series are series of inverses of integers whose expansion in base $B$, for some $B \geq 2$, contains no occurrence of some fixed digit or some fixed block of digits. A prototypical example was proposed by Kempner in 1914, namely the sum inverses of integers whose expansion in base $10$ contains no occurrence of a nonzero given digit. Results about such series address their convergence as well as closed expressions for their sums (or approximations thereof). Another direction of research is the study of sums of inverses of integers that contain only a given finite number, say $k$, of some digit or some block of digits, and the limits of such sums when $k$ goes to infinity. Generalizing partial results in the literature, we give a complete result for any digit or block of digits in any base.

math.NT↗

arXiv2023

Kempner-like harmonic series

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 $\lim_{k \to \infty}\sum_{a(n) = k} 1/n = 4 \log 2$.

math.NT↗