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Integers with an average sum of digits. (Sur les entiers dont la somme des chiffres est moyenne.) (French) Zbl 1084.11045

Let \(s(n)\) denote the sum of digits in base \(g\). The authors prove a local limit law for this function in the following strange form. Write \(g_1=(g-1)/2\) and let \(b\) be a function such that \(b(n)=O(n^{1/4})\) and \(g_1n + b(n)\) is always integer. Then the number of integers \(n\leq x\) such that \[ s(n) = g_1 [ \log _g n] + b([ \log _g n]) \eqno{(*)} \] is \( c x ( \log _g x )^{-1/2} + O(x/ \log x )\) with a constant \(c\). They also show that for every irrational \(\alpha \) the sequence \(\alpha n\), where \(n\) runs over the integers satisfying \((*)\), is uniformly distributed modulo 1.
The reviewer does not see any reason why the range of \(b\) is so restricted; one would expect a central limiting law for \(b(n) \sim t \sqrt { \log n}\) with perhaps a weaker remainder term.

MSC:

11K65 Arithmetic functions in probabilistic number theory
11A63 Radix representation; digital problems
11N37 Asymptotic results on arithmetic functions

Keywords:

sum of digits
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References:

[1] Bush, L. E., An asymptotic formula for the average sum of the digits of integers, Amer. Math. Monthly, 47, 154-156 (1940) · Zbl 0025.10601
[2] Coquet, J., Sur certaines suites uniformément équiréparties modulo 1, Acta Arith., 36, 157-162 (1980) · Zbl 0357.10026
[3] Delange, H., Sur la fonction sommatoire de la fonction “somme des chiffres”, Enseign. Math., 21, 31-47 (1975) · Zbl 0306.10005
[4] Mauduit, C.; Sárközy, A., On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith., 81, 145-173 (1997) · Zbl 0887.11008
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