arXiv subjects
Publications and source records attributed to Élie Goudout.
Let $x\geqslant 3$. For $1\leqslant n\leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that $ω(n-1)$ satisfies an Erdős-Kac type theorem whenever $ω(n)=k$ where $1\leqslant k\ll\log\log x$, thus extending a result of Halberstam.
We study the simultaneous concentration of the values of several additive functions along polynomial shifts. Under a slight restriction, this yields an extension of a result from Halász in 1975.
For $x\geq 3$, we define $w(x)$ as the highest integer $w$ for which there exist integers $m, y\geq 1$ and $1\leq n_1<\dots<n_m\leq x$ such that $n_1\cdots n_m=y^w$. We show that \[w(x)=x\exp\big(-(\sqrt{2}+o(1))\sqrt{\log x\log\log x}\big).\]
For $k\geq 1$ an integer and $x\geq 1$ a real number, let $π_k(x)$ be the number of integers smaller than $x$ having exactly $k$ distinct prime divisors. Building on recent work of Matomäki and Radziwiłł, we investigate the asymptotic behavior of $π_k(x+h)-π_k(x)$ for almost all $x$, when $h$ is very small. We obtain optimal results for $k\asymp\log_2 x$ and close to optimal results for $5\leq k\leq\log_2 x$. Our method also applies to $y$-friable integers in almost all intervals $[x,x+h]$ when $\frac{\log x}{\log y}\leq (\log x)^{1/6-\varepsilon}$.
We show that the Erdős-Kac theorem is valid in almost all intervals $\left[x,x+h\right]$ as soon as $h$ tends to infinity with $x$. We also show that for all $k$ near $\log\log x$, almost all interval $\left[x,x+\exp\left(\left(\log\log x\right)^{1/2+\varepsilon}\right)\right]$ contains the expected number of integers $n$ such that $ω(n)=k$. These results are a consequence of the methods introduced by Matomäki and Radziwiłł to estimate sums of multiplicative functions over short intervals.