Applications of binomial measures to power sums of digital sums
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.
scientific article; zbMATH DE number 770121