The error term of the sum of digital sum functions in arbitrary bases

Erdenebileg Erdenebat and Ka Lun Wong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 311–318
DOI: 10.7546/nntdm.2024.30.2.311-318
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Authors and affiliations

Erdenebileg Erdenebat
Faculty of Math and Computing, Brigham Young University–Hawaii
55-220 Kulanui Street, Laie, HI 96762, USA

Ka Lun Wong
Faculty of Math and Computing, Brigham Young University–Hawaii
55-220 Kulanui Street, Laie, HI 96762, USA

Abstract

Let k be a non-negative integer and q > 1 be a positive integer. Let s_q(k) be the sum of digits of k written in base q. In 1940, Bush proved that A_q(x)=\sum_{k \leq x} s_q (k) is asymptotic to \frac{q-1}{2}x \log_q x. In 1968, Trollope proved an explicit formula for the error term of A_2(n-1), labeled by -E_2(n), where n is a positive integer. In 1975, Delange extended Trollope’s result to an arbitrary base q by another method and labeled the error term nF_q(\log_q n). When q=2, the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope’s method to extend -E_2(n) to -E_q(n) with a general base q, and we also proved directly that nF_q(\log_q n) and -E_q(n) are equal for any q.

Keywords

  • Digital sums
  • Asymptotic
  • Error term

2020 Mathematics Subject Classification

  • 11A25
  • 11A63
  • 11N37

References

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Manuscript history

  • Received: 30 August 2023
  • Revised: 2 May 2024
  • Accepted: 13 May 2024
  • Online First: 19 May 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Erdenebat, E., & Wong, K. L. (2024). The error term of the sum of digital sum functions in arbitrary bases. Notes on Number Theory and Discrete Mathematics, 30(2), 311-318, DOI: 10.7546/nntdm.2024.30.2.311-318.

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