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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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