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
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Authors and affiliations

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


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.


  • Binary digital sum
  • Dirichlet convolution

2010 Mathematics Subject Classification

  • 11A63


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Cite this paper

Srichan, T. (2019). Averages of the Dirichlet convolution of the binary digital sum. Notes on Number Theory and Discrete Mathematics, 25(1), 122-127, doi: 10.7546/nntdm.2019.25.1.122-127.

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