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arXiv · 2608.14241

Ratio of sum of digits functions in two bases

Pascal Jelinek
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Abstract

In 2019 La Bret\`eche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of $\mathbb{N}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.

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Pascal Jelinek. 2026-08-14. Ratio of sum of digits functions in two bases. https://arxiv.org/abs/2608.14241

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