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Sums of digits of multiples of integers. (Sommes des chiffres de multiples d’entiers.) (French) Zbl 1110.11025
For a positive integer \(n\), let \(s_q(n)\) denote the sum of digits of \(n\) when it is written in base \(q\) and the vector function \(s_q({\mathbf h}n):=(s_q(h_1n),\dots ,s_q(h_rn))\), where \({\mathbf h}:=(h_1,\dots ,h_r)\in(\mathbb{N}^*)^r\). A number of results has been obtained for the function \(s_q({\mathbf h}n)\) when the vector h is fixed. For applications, a control of dependence on the parameter h is required. The paper under review is devoted to study this subject. In particular the authors are interested on the sums
\[ \sum_{ n\leq x}e({\mathbf v}.s_q({\mathbf h}n) +\theta n,\quad \sum_{x< n\leq x+y}e({\mathbf v}.s_q({\mathbf h}n) +\theta n)\quad \text{and}\quad \sum_{x<\leq x+y}e({\mathbf v}.s_q({\mathbf h}n+{\mathbf k}) +\theta n), \] where \({{\mathbf v}=(v_1,\dots ,v_r)\in \mathbb{R}^r}\), \({{\mathbf k}=(k_1,\dots ,k_r)\in\mathbb{N}^r}\), \(\theta\in \mathbb{R}\), \({\mathbf v}.s_q({\mathbf h}n+{\mathbf k}):= \sum_{1\leq j\leq r}v_js_q(h_jn+k_j)\) and as usual \(e(a):= \exp(2\pi ia)\). The basic theorems of this work are technical and too complicated to be stated in this review. The paper also contains interesting applications; we mention here the three main ones: The first is about Gel’fond’s conjecture. In his paper [Acta Arith. 13, 259–265 (1968; Zbl 0155.09003)] A. O. Gel’fond conjectured that
\[ \sum_{\substack{ p\leq x \\ s_q(p)\equiv a\pmod m }} 1\sim\frac x{m\log x}\quad ((m,q-1)=1,\;x\rightarrow\infty). \] By using sieve methods, E. Fouvry and C. Mauduit proved [Acta Arith. 77, No. 4, 339–351 (1996; Zbl 0869.11073)] that
\[ \sum _{\substack{ n\leq x, n=p \text{ or } n=p_1p_2 \\ s_q(p)\equiv a\pmod m }} 1\gg\frac x{\log x}\;, \] provided \((m,q-1)=1\). Under the same hypothesis, the authors of the present paper establish
\[ \sum_{\substack{ n\in E_k(x)\\s_q(n)\equiv a\pmod m}} 1\gg_{k,m,q} \frac{x(\log_2x)^{k-2}}{(\log_3x)\log x}\;, \] where \( E_k(x)\) denotes the set of positive integers \(n\leq x\) having exactly \(k\) prime factors.
The second application is an explicit asymptotic equivalent of \[ A(x,{\mathbf h},{\mathbf a},{\mathbf m},b,d):= \sum_{\substack{ n\leq x, n\equiv b\pmod d \\ s_q(h_jn)\equiv a_j\pmod m_j\quad 1\leq j\leq r }} 1, \] where \({\mathbf a}=(a_1,\dots ,a_j)\) and \({\mathbf m}=(m_1,\dots ,m_j)\) are in \((\mathbb{N}^*)^r\). In this subject, the authors generalize and extend a result of J. A. Solinas [J. Number Theory 33, No. 2, 132–151 (1989; Zbl 0678.10037)] and by using a large sieve inequality they obtain a statistical estimate of the Bombieri-Vinogradov type:
\[ \sum_{ d\leq D,\; (q,d)=1} \left| A(x,{{\mathbf h}},{{\mathbf a}},{{\mathbf m}},b,d)- \frac{x}{m_1\dots m_r d}\right| \ll_{\beta,{{\mathbf m}},q}\frac{x}{(\log x)^{\beta}}\;, \] where \(\beta\) is a fixed real positif number and \(D=\sqrt{x}/(\log x)^{\beta+2}\).
The last application, which we give in this review, is an analogue of the Daboussi-Delange theorem. Let \(f\) a multiplicative function satisfies \({\sum_{n\leq x}| f(n)| ^2=O(x)}\) and \(\vartheta\) an irrational real number, then H. Daboussi and H. Delange [J. Lond. Math. Soc. (2) 26, No. 2, 245–264 (1982; Zbl 0499.10052)] proved that we have \[ \lim_{x\rightarrow\infty}\frac1x \sum_{n\leq x}f(n)e(\vartheta n)=0. \] Under some hypothesis on the vectors \({\mathbf v}\) , \({\mathbf h}\) and the integer \(q\), the authors prove that \[ \sum_{n\leq x}f(n)e({{\mathbf v}}.s_q({{\mathbf h}}n))\ll\frac x{\log_2x}\;, \] when \(f\) is a complex multiplicative function of modulus at most 1.

MSC:
11L07 Estimates on exponential sums
11B85 Automata sequences
11A63 Radix representation; digital problems
11N36 Applications of sieve methods
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