**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

**Full paper (PDF, 219 Kb)**

## Details

### 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 be a non-negative integer and be a positive integer. Let be the sum of digits of written in base In 1940, Bush proved that is asymptotic to In 1968, Trollope proved an explicit formula for the error term of labeled by where is a positive integer. In 1975, Delange extended Trollope’s result to an arbitrary base by another method and labeled the error term When 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 to with a general base and we also proved directly that and are equal for any

### 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.