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Congruences of sums of digits of polynomial values. (Congruences de sommes de chiffres de valeurs polynomiales.) (French) Zbl 1153.11307
Authors’ summary: Let \(m, g, q\in \mathbb N\) with \(q \geq 2\) and \((m, q-1) = 1\). For \(n\in \mathbb N\), denote by \(s_n(n)\) the sum of digits of \(n\) in the \(q\)-ary digital expansion. Given a polynomial \(f\) with integer coefficients, degree \(d\geq 1\), and such that \(f(N) \subset \mathbb N\), it is shown that there exists \(C = C(f, m, q) > 0\) such that for any \(g\in \mathbb Z\), and all large \(N\),
\[ |\{0\leq n\leq N: s_q(f(n))\equiv g\pmod m\}|\geq CN^{\min(1,2/d!)}. \]
In the special case \(m = q = 2\) and \(f(n) = n^2\), the value \(C = 1/20\) is admissible.

MSC:
11B85 Automata sequences
11N37 Asymptotic results on arithmetic functions
11N69 Distribution of integers in special residue classes
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