On the distribution of the truncated sum-of-digits function of polynomial sequences in residue classes
Let \(q\ge 2\) be an integer and let \(s_q(n)\) denote the sum of digits of the positive integer \(n\) in base \(q.\) The properties of \(s_q(n)\) on polynomial sequences have been studied by many authors. E.g., for \(P(x) \in \mathbb{Z}[x],\) the normality theorem of \textit{H. Davenport} and \textit{P. Erdős} [Can. J. Math. 4, 58--63 (1952; Zbl 0046.04902)] implies that \[\sum_{n\le x} s_q(P(n)) \sim \frac {q-1}{2} \deg (P)\;x \log_q x\] as \(x\to \infty.\) \textit{N. L. Bassily} and \textit{I. Kátai} [Acta Math. Hung. 68, 353--361 (1995; Zbl 0832.11035)] showed a central limit theorem for \(s_q(P(n)).\) \textit{A. O. Gelfond} [Acta Arith. 13, 259--265 (1968; Zbl 0155.09003)] studied the distribution of \(s_q(n)\) in residue classes and also suggested to evaluate \[ \sum_{\substack{n\le x \\ s_q(P(n))\equiv a \bmod m}} 1 \] \textit{C. Dartyge} and \textit{G. Tenenbaum} [Bull. Lond. Math. Soc. 38, 61--69 (2006; Zbl 1153.11307)] gave a first answer to Gelfond's problem. Their lower bound was improved by \textit{T. Stoll} [Funct. Approx., Comment. Math. 47, 233--239 (2012; Zbl 1315.11010)]. \textit{C. Mauduit} and \textit{J. Rivat} [Acta Math. 203, 107--148 (2009; Zbl 1278.11076)] gave a precise answer to Gelfond's problem for \(P(n)=n^2\). This result was then generalized by \textit{M. Drmota,} \textit{C. Mauduit,} and \textit{J. Rivat} [J. Lond. Math. Soc., II. Ser. 84, 81--102 (2011; Zbl 1257.11009)] for prime \(q\) large enough in terms of \(\deg (P)\). In the paper under review the authors consider the truncated sum-of-digits function \[(s_q(P(n)\bmod ; q^k))_{n < q^k}\] introduced by \textit{M. Drmota} and \textit{J. Rivat} [J. Lond. Math. Soc., II. Ser. 72, 273--292 (2005; Zbl 1092.11006)]. Theorem 1.1 of the paper under review states an asymptotic formula for the distribution of \[(s_q(P(n)\bmod q^k))_{n < q^k}\] in residue classes modulo \(m\) for any integers \(k,q\ge 2\) by using the method introduced by \textit{C. Mauduit} and \textit{J. Rivat} [ibid.]. Theorem 1.2 shows asymptotic formulae for the distribution of \[(s_p(n^d\bmod p^k))_{n<p^k}\] in residue classes modulo \(m\) for any integers \(k,d\ge 2\) and any prime \(p>2\) by introducing a new approach based on Dirichlet character sums. Theorem 1.3 states exact identities for \(m=p, \;(d, p)=1.\)
scientific article; zbMATH DE number 7377594
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