Skip to main content
Log in

The sum-of-digits function in rings of residue classes

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Dartyge and Sárközy introduced the notion of digits in finite fields and studied the properties of polynomial values of \({\mathbb {F}}_q\) with a fixed sum of digits. Swaenepoel provided sharp estimates for the number of elements of special sequences of \({\mathbb {F}}_q\) whose sum of digits is prescribed. In this paper we study the sum-of-digits function in rings of residue classes and give a few asymptotic formulas and exact identities by using estimates for character sums and exponential sums modulo prime powers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. K. Aloui, C. Mauduit, M. Mkaouar, On the correlation of the sum of digits along prime numbers. Acta Arith. 199(3), 275–301 (2021)

    MathSciNet  Google Scholar 

  2. J.H.H. Chalk, On Hua’s estimate for exponential sums. Mathematika 34(2), 115–123 (1987)

    Article  MathSciNet  Google Scholar 

  3. J.R. Chen, On Professor Hua’s estimate of exponential sums. Sci. Sinica 20(6), 711–719 (1977)

    MathSciNet  Google Scholar 

  4. T. Cochrane, Exponential sums modulo prime powers. Acta Arith. 101(2), 131–149 (2002)

    Article  MathSciNet  Google Scholar 

  5. T. Cochrane, Z. Zheng, Pure and mixed exponential sums. Acta Arith. 91(3), 249–278 (1999)

    Article  MathSciNet  Google Scholar 

  6. T. Cochrane, Z. Zheng, Exponential sums with rational function entries. Acta Arith. 95(1), 67–95 (2000)

    Article  MathSciNet  Google Scholar 

  7. T. Cochrane, Z. Zheng, On upper bounds of Chalk and Hua for exponential sums. Proc. Amer. Math. Soc. 129(9), 2505–2516 (2001)

    Article  MathSciNet  Google Scholar 

  8. A.H. Copeland, P. Erdős, Note on normal numbers. Bull. Amer. Math. Soc. 52, 857–860 (1946)

    Article  MathSciNet  Google Scholar 

  9. C. Dartyge, A. Sárközy, The sum of digits function in finite fields. Proc. Amer. Math. Soc. 141(12), 4119–4124 (2013)

    Article  MathSciNet  Google Scholar 

  10. C. Dartyge, G. Tenenbaum, Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble) 55(7), 2423–2474 (2005)

    Article  MathSciNet  Google Scholar 

  11. H. Davenport, P. Erdős, Note on normal decimals. Canadian J. Math. 4, 58–63 (1952)

    Article  MathSciNet  Google Scholar 

  12. H. Delange, Sur la fonction sommatoire de la fonction somme des chiffres. Enseign. Math. (2) 21(1), 31–47 (1975)

    MathSciNet  Google Scholar 

  13. P. Ding, An improvement to Chalk’s estimation of exponential sums. Acta Arith. 59(2), 149–155 (1991)

    Article  MathSciNet  Google Scholar 

  14. P. Ding, On a conjecture of Chalk. J. Number Theory 65(1), 116–129 (1997)

    Article  MathSciNet  Google Scholar 

  15. M. Drmota, J. Rivat, The sum-of-digits function of squares. J. London Math. Soc. (2) 72(2), 273–292 (2005)

    Article  MathSciNet  Google Scholar 

  16. M. Drmota, C. Mauduit, J. Rivat, Primes with an average sum of digits. Compos. Math. 145(2), 271–292 (2009)

    Article  MathSciNet  Google Scholar 

  17. M. Drmota, C. Mauduit, J. Rivat, The sum-of-digits function of polynomial sequences. J. Lond. Math. Soc. (2) 84(1), 81–102 (2011)

    Article  MathSciNet  Google Scholar 

  18. E. Fouvry, C. Mauduit, Sur les entiers dont la somme des chiffres est moyenne. J. Number Theory 114(1), 135–152 (2005)

    Article  MathSciNet  Google Scholar 

  19. A.O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13, 259–265 (1967/68)

  20. L.K. Hua, On an exponential sum. J. Chinese Math. Soc. 2, 301–312 (1940)

    MathSciNet  Google Scholar 

  21. L.K. Hua, Introduction to number theory (Translated from the Chinese by Peter Shiu. Springer Verlag, Berlin, 1982)

    Google Scholar 

  22. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Oxford Science Publications, Oxford, 1979)

    Google Scholar 

  23. H. Liu, C. Mauduit, On the distribution of the truncated sum-of-digits function of polynomial sequences in residue classes. Acta Math. Hungar. 164(2), 360–376 (2021)

    Article  MathSciNet  Google Scholar 

  24. W.K.A. Loh, On Hua’s lemma. Bull. Austral. Math. Soc. 50(3), 451–458 (1994)

    Article  MathSciNet  Google Scholar 

  25. M. Lu, Estimate of a complete trigonometric sum. Sci. Sinica Ser. A 28(6), 561–578 (1985)

    MathSciNet  Google Scholar 

  26. C. Mauduit, J. Rivat, La somme des chiffres des carrés. Acta Math. 203(1), 107–148 (2009)

    Article  MathSciNet  Google Scholar 

  27. C. Mauduit, J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Ann. of Math. (2) 171(3), 1591–1646 (2010)

    Article  MathSciNet  Google Scholar 

  28. C. Mauduit, J. Rivat, Prime numbers along Rudin-Shapiro sequences. J. Eur. Math. Soc. (JEMS) 17(10), 2595–2642 (2015)

    Article  MathSciNet  Google Scholar 

  29. C. Mauduit, J. Rivat, Rudin-Shapiro sequences along squares. Trans. Amer. Math. Soc. 370(11), 7899–7921 (2018)

    Article  MathSciNet  Google Scholar 

  30. C. Mauduit, A. Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed. Acta Arith. 81(2), 145–173 (1997)

    Article  MathSciNet  Google Scholar 

  31. M. Peter, The summatory function of the sum-of-digits function on polynomial sequences. Acta Arith. 104(1), 85–96 (2002)

    Article  MathSciNet  Google Scholar 

  32. I. Shiokawa, On the sum of digits of prime numbers. Proc. Japan Acad. 50, 551–554 (1974)

    MathSciNet  Google Scholar 

  33. C. Swaenepoel, On the sum of digits of special sequences in finite fields. Monatsh. Math. 187(4), 705–728 (2018)

    Article  MathSciNet  Google Scholar 

  34. W. Zhang, W. Yao, A note on the Dirichlet characters of polynomials. Acta Arith. 115(3), 225–229 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors express their gratitude to the referee for his/her helpful and detailed comments. This paper is supported by the National Natural Science Foundation of China under Grant No. 12071368, and the Shaanxi Fundamental Science Research Project for Mathematics and Physics under Grant No. 22JSY017.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zehua Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Liu, Z. The sum-of-digits function in rings of residue classes. Period Math Hung 89, 335–354 (2024). https://doi.org/10.1007/s10998-024-00595-0

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-024-00595-0

Keywords

Mathematics Subject Classification

Navigation