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The truncated sum-of-digits function of powers

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Abstract

Let q ≥ 2 be an integer and let \(s_{q}(n)\) be the sum-of-digitsfunction of n in base q. The function \(s_{q}(n)\) has been studied in many directions and many properties have been obtained on the distribution of \(s_{q}(n)\) and \(s_{q}(P(n))\), where P is a suitable polynomial. In this paper we derive the generatingfunctions of \(s_{p}(n^{d} {\rm mod} p^{k})\) for \(d\geq 2\) and prime \(p\geq 2\) by using the properties of Dirichlet character, and study the correlation properties of the sequences \(((-1)^{n^2 \bmod p^k})_{n<p^k}\).

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Authors and Affiliations

  1. Research Center for Number Theory and its Applications, School of Mathematics, Northwest University, Xi’an, 710127, China

    H. Liu & Y. Qi

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  1. H. Liu
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  2. Y. Qi
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Correspondence to Y. Qi.

Additional information

This work is supported by National Natural Science Foundation of China under Grant No. 12071368, and the Science and Technology Program of Shaanxi Province of China under Grants No. 2019JM-573 and 2020JM-026.

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Liu, H., Qi, Y. The truncated sum-of-digits function of powers. Acta Math. Hungar. 168, 27–49 (2022). https://doi.org/10.1007/s10474-022-01267-6

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  • DOI: https://doi.org/10.1007/s10474-022-01267-6

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