Applications of binomial measures to power sums of digital sums
type: Item
instance of: scholarly article
title: Applications of binomial measures to power sums of digital sums
zbMATH Open document ID: 0824.11004
author: Tatsuya Okada
author: Takeshi Sekiguchi
author: Yasunobu Shiota
published in: Journal of Number Theory
publication date: 1995-11-15T00:00:00Z
review text: The paper is devoted to the asymptotic study of the digital sum \(S_ p (N)= \sum_{n<N} s(n)^ p\), where \(s(n)\) denotes the sum of binary digits of \(n\) and \(p\) is a given positive integer. \textit{J. Coquet} [J. Number Theory 22, 161-176 (1986; Zbl 0578.10009)] obtained an explicit formula of the type \[ S_ p (N)= N\Biggl( {{\log_ 2 N} \over 2}\Biggr)^ p+ N\sum_{0\leq k<p} (\log_ 2 N)^ k G_{p,k} (\log_ 2 N), \] where the periodic functions \(G_{p,k}\) satisfy a certain recurrence relation. Furthermore, Coquet conjectured the continuity of the periodic functions \(G_{p,k}\). The authors apply the binomial measure to prove this conjecture and to obtain further explicit formulas on the functions \(G_{p,k}\). The result is a precise analogon of the well-known Delange-Trollope formula.
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 11K16
zbMATH DE Number: 770121
zbMATH Keywords: digital sum
zbMATH Keywords: binomial measure
zbMATH Keywords: asymptotic study
zbMATH Keywords: Delange-Trollope formula
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
full work available at URL: jnth.1995.1068
OpenAlex ID: W2007342124
DOI: 10.1006/JNTH.1995.1068
description: scientific article; zbMATH DE number 770121