The distribution of numbers with many ordered factorizations

Lebowitz-Lockard, Noah

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 583-606.

Let $g\left(n\right)$ be the number of ordered factorizations of $n$ into numbers larger than $1$. We find precise bounds on the positive moments of $g$. We use these results to estimate the number of $n\le x$ satisfying $g\left(n\right)\ge x^{\alpha }$ for all positive $\alpha$. In addition, let $G\left(n\right)$ and $g_{𝒫}\left(n\right)$ be the number of ordered factorizations of $n$ into distinct numbers larger than $1$ and primes, respectively. We also bound the positive moments of $G$ and $g_{𝒫}$ from below.

Soit $g\left(n\right)$ le nombre de factorisations de $n$ en produit ordonné de facteurs plus grands que $1$. On trouve des bornes précises pour les moments positifs de $g$. On utilise ces résultats pour estimer le nombre de $n\le x$ tels que $g\left(n\right)\ge x^{\alpha }$ pour tous les $\alpha$ positifs. En outre, soient $G\left(n\right)$ et $g_{𝒫}\left(n\right)$ les nombres de factorisations de $n$ en produit ordonné de facteurs distincts plus grands que $1$ et en produit ordonné de facteurs premiers respectivement. On donne des bornes inférieures pour les moments positifs de $G$ et $g_{𝒫}$.

Classification: 11A25, 11A51, 11N37
Keywords: Ordered factorizations

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