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.
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References
K. Aloui, C. Mauduit, M. Mkaouar, On the correlation of the sum of digits along prime numbers. Acta Arith. 199(3), 275–301 (2021)
J.H.H. Chalk, On Hua’s estimate for exponential sums. Mathematika 34(2), 115–123 (1987)
J.R. Chen, On Professor Hua’s estimate of exponential sums. Sci. Sinica 20(6), 711–719 (1977)
T. Cochrane, Exponential sums modulo prime powers. Acta Arith. 101(2), 131–149 (2002)
T. Cochrane, Z. Zheng, Pure and mixed exponential sums. Acta Arith. 91(3), 249–278 (1999)
T. Cochrane, Z. Zheng, Exponential sums with rational function entries. Acta Arith. 95(1), 67–95 (2000)
T. Cochrane, Z. Zheng, On upper bounds of Chalk and Hua for exponential sums. Proc. Amer. Math. Soc. 129(9), 2505–2516 (2001)
A.H. Copeland, P. Erdős, Note on normal numbers. Bull. Amer. Math. Soc. 52, 857–860 (1946)
C. Dartyge, A. Sárközy, The sum of digits function in finite fields. Proc. Amer. Math. Soc. 141(12), 4119–4124 (2013)
C. Dartyge, G. Tenenbaum, Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble) 55(7), 2423–2474 (2005)
H. Davenport, P. Erdős, Note on normal decimals. Canadian J. Math. 4, 58–63 (1952)
H. Delange, Sur la fonction sommatoire de la fonction somme des chiffres. Enseign. Math. (2) 21(1), 31–47 (1975)
P. Ding, An improvement to Chalk’s estimation of exponential sums. Acta Arith. 59(2), 149–155 (1991)
P. Ding, On a conjecture of Chalk. J. Number Theory 65(1), 116–129 (1997)
M. Drmota, J. Rivat, The sum-of-digits function of squares. J. London Math. Soc. (2) 72(2), 273–292 (2005)
M. Drmota, C. Mauduit, J. Rivat, Primes with an average sum of digits. Compos. Math. 145(2), 271–292 (2009)
M. Drmota, C. Mauduit, J. Rivat, The sum-of-digits function of polynomial sequences. J. Lond. Math. Soc. (2) 84(1), 81–102 (2011)
E. Fouvry, C. Mauduit, Sur les entiers dont la somme des chiffres est moyenne. J. Number Theory 114(1), 135–152 (2005)
A.O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13, 259–265 (1967/68)
L.K. Hua, On an exponential sum. J. Chinese Math. Soc. 2, 301–312 (1940)
L.K. Hua, Introduction to number theory (Translated from the Chinese by Peter Shiu. Springer Verlag, Berlin, 1982)
G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Oxford Science Publications, Oxford, 1979)
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)
W.K.A. Loh, On Hua’s lemma. Bull. Austral. Math. Soc. 50(3), 451–458 (1994)
M. Lu, Estimate of a complete trigonometric sum. Sci. Sinica Ser. A 28(6), 561–578 (1985)
C. Mauduit, J. Rivat, La somme des chiffres des carrés. Acta Math. 203(1), 107–148 (2009)
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)
C. Mauduit, J. Rivat, Prime numbers along Rudin-Shapiro sequences. J. Eur. Math. Soc. (JEMS) 17(10), 2595–2642 (2015)
C. Mauduit, J. Rivat, Rudin-Shapiro sequences along squares. Trans. Amer. Math. Soc. 370(11), 7899–7921 (2018)
C. Mauduit, A. Sárközy, On the arithmetic structure of the integers whose sum of digits is fixed. Acta Arith. 81(2), 145–173 (1997)
M. Peter, The summatory function of the sum-of-digits function on polynomial sequences. Acta Arith. 104(1), 85–96 (2002)
I. Shiokawa, On the sum of digits of prime numbers. Proc. Japan Acad. 50, 551–554 (1974)
C. Swaenepoel, On the sum of digits of special sequences in finite fields. Monatsh. Math. 187(4), 705–728 (2018)
W. Zhang, W. Yao, A note on the Dirichlet characters of polynomials. Acta Arith. 115(3), 225–229 (2004)
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.
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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
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DOI: https://doi.org/10.1007/s10998-024-00595-0