This is an account on two joint works with Cécile Dartyge (Nancy 1).Let with and . For , we denote by the sum of digits of in the -ary digital expansion. Let be a quadratic polynomial with integer coefficients. We show that there exists such that for all , ,
While a fairly simple proof of (1) may be provided when , the general case requires rather sophisticated tools. The approach followed in the above-mentioned works is based on an upper bound for the quantity
Other applications of this general device will be described, in particular the following two, where congruence properties of sums of digits may be combined with multiplicative constraints.
(i) Given positive integers with , , , there exist infinitely many integers having exactly prime factors and such that . This improves on a result of Étienne Fouvry and Christian Mauduit by removing an indetermination arising from the so-called parity phenomenon.
(ii) Daboussi-type theorem for sums of digits.
Let , , and . Uniformly for
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