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PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 1 / January 2019
- Published online by Cambridge University Press:
- 04 April 2017, pp. 189-224
- Print publication:
- January 2019
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Let
$b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum where the summation runs over prime numbers
$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$
$p$ and where 
$\unicode[STIX]{x1D6FD}\in \mathbb{R}$, 
$k\in \mathbb{Z}$, and 
$g:\mathbb{N}\rightarrow \mathbb{Z}$ is a strongly 
$b$-additive function such that 
$\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$.
STRONGLY
$q$-ADDITIVE FUNCTIONS AND DISTRIBUTIONAL PROPERTIES OF THE LARGEST PRIME FACTOR
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 17 November 2015, pp. 177-185
- Print publication:
- April 2016
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- Article
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Let
$P(n)$ denote the largest prime factor of an integer 
$n\geq 2$. In this paper, we study the distribution of the sequence 
$\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer 
$b\geq 2$, where 
$f$ is a strongly 
$q$-additive integer-valued function (that is, 
$f(aq^{j}+b)=f(a)+f(b),$ with 
$(a,b,j)\in \mathbb{N}^{3}$, 
$0\leq b<q^{j}$). We also show that the sequence 
$\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if 
${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.
