Ratio of sum of digits functions in two bases
Abstract
In 2019 La Bretèche, Stoll and Tennenbaum showed that the ratio of the sum of digits function sq1(n)/sq2(n) of two multiplicatively independent bases q1 and q2 is dense in ℚ+. Recently Spiegelhofer proved that in the special case s2(n)/s3(n)=1 we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair (s2(n),s3(n)) attains almost every value of ℕ2 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.
Related
- On Integer-Division Bases for the Kalmár Elementary Functions
- Convexity of a ratio function involving polygamma functions and monotonicity of the sectional curvature of the Fisher--Rao manifold
- Reduction of integer tiles via CRT and base-p digits
- Irreducible polynomials with restricted digits in base b(T)
- On the digits of the sum of proper divisors
BibTeX
@misc{jelinek2026ratio,
title = {{Ratio of sum of digits functions in two bases}},
author = {Pascal Jelinek},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.14241},
eprint = {2608.14241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14241},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ratio of sum of digits functions in two bases
AU - Pascal Jelinek
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.14241
UR - https://arxiv.org/abs/2608.14241
AB - In 2019 La Bretèche, 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 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 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.
ER -