Averages of the Dirichlet convolution of the binary digital sum

Teerapat Srichan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 122–127
DOI: 10.7546/nntdm.2019.25.1.122-127
Full paper (PDF, 144 Kb)

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Authors and affiliations

Teerapat Srichan
Department of Mathematics, Faculty of Science
Kasetsart University, Bangkok, Thailand

Abstract

We derive some averages of the Dirichlet convolution of the binary digital sum s₂(n), the sum of digits ofthe expansion of n in base 2. The Trollope–Delange formula is used in our proof. It provides an explicit asymptotic formula for the total number of digits ‘1’ in the binary expansions of the integers between 1 and n − 1 in term of the continuous, nowhere differentiable Takagi function. Moreover, we also extend the result to averages of the k-th convolution of the binary digital sum, for k ≥ 2.

Keywords

• Binary digital sum
• Dirichlet convolution

2010 Mathematics Subject Classification

• 11A63

References

1. Bellman, R., & Shapiro, H. N. (1948). On a problem in additive number theory. Annals of Mathematics, 49 (2), 333–340.
2. Coquet, J. (1986). Power sums of digital sums. Journal of Number Theory, 22 (2), 161–176.
3. Delange, H. (1975). Sur la fonction sommatoire de la fonction “somme” des chiffres. Enseignement Math.(2), 21 (1), 31–47.
4. Grabner, P. J., Kirschenhofer, P., Prodinger, H., & Tichy, R. F. (1993). On the moments of the sum-of-digits function. Applications of Fibonacci numbers, Springer, Dordrecht, 263–271.
5. Krüppel, M. (2007). On the extrema and the improper derivatives of Takagi’s continuous nowhere differentiable function, Rostock. Math. Kolloq, 62, 41–59.
6. Mirsky, L. (1949). A theorem on representations of integers in the scale of . Scripta Math, 15, 11–12.
7. Sándor, J., & Crstici, B. (2004). Handbook of Number Theory II., Kluwer Academic, Dordrecht.
8. Trollope, J. R. (1968). An explicit expression for binary digital sums. Mathematics Magazine, 41 (1), 21–25. .