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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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##### References:
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