Gerald Tenenbaum: Sums of digits of polynomial values

This is a joint work with CÚcile Dartyge.

This is an account on two joint works with Cécile Dartyge (Nancy 1).

Let $ m,q \in{\mathbb{N}}$ with $ q\geqslant 2$ and $ (m,q-1)=1$. For $ n\in{\mathbb{N}}$, we denote by $ s_q(n)$ the sum of digits of $ n$ in the $ q$-ary digital expansion. Let $ f\in{\mathbb{Z}}[X]$ be a quadratic polynomial with integer coefficients. We show that there exists $ C=C(f,m,q)>0$ such that for all $ N\geqslant 1$, $ k\in{\mathbb{Z}}$,

$\displaystyle \vert\{ n\leqslant N : s_q(f(n))\equiv k ({\rm mod }m)\}\vert\geqslant CN.\leqno{(1)}$

In the special case $ m=q=2$ and $ f(n)=n^2$, the value $ C=1/20$ is admissible.

While a fairly simple proof of (1) may be provided when $ m=2$, the general case requires rather sophisticated tools. The approach followed in the above-mentioned works is based on an upper bound for the quantity

$\displaystyle G(x,y):=\sum_{x<n\leqslant x+y}{\rm e}\Big (\sum_{j=1}^r \alpha _j
s_q(h_jn)\Big )$

when $ \alpha_1 ,\ldots ,\alpha_r\in {\mathbb{R}}$, $ h_1,\ldots ,h_r\in {\mathbb{N}}$, $ x\geqslant 1$, $ y\geqslant 1$, with the usual notation $ {\rm e}(u):={\rm e}^{2\pi iu}$.

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 $ a,k,m,q$ with $ k\geqslant 2$, $ q\geqslant 2$, $ (m,q-1)=1$, there exist infinitely many integers $ n$ having exactly $ k$ prime factors and such that $ s_q(n)\equiv a ({\rm mod }m)$. 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 $ q\in{\mathbb{N}}^*$, $ c\in]0,1/9[$, and $ \alpha_1, \alpha_2\in{\mathbb{R}}^2\smallsetminus\big\{{\mathbb{Z}}/(q-1)\big\}^2$. Uniformly for

$\displaystyle x>2,\quad h_1, h_2\in{\mathbb{N}}^*,\quad 0<h_1< h_2 <(\log x)^{c},\quad
q\nmid h_2,$

and multiplicative complex arithmetic function $ f$ with values in the unit disc, we have

$\displaystyle \sum_{n\leq x}{\rm e}\Big (\alpha_1s_q(h_1 n)+\alpha_2s_q(h_2n)\Big)f(n)\ll
\frac{x}{\log\log x}.\leqno(2)$

This is stated in dimension 2 for simplicity, but the result may be extended to any finite dimension.

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