On the unimodal character of the frequency function of the largest prime factor

Jean-Marie De Koninck¹ ; Jason Pierre Sweeney¹

¹ Département de Mathématiques et de Statistique Université Laval Québec G1K 7P4, Canada

Colloquium Mathematicum, Tome 88 (2001) no. 2, pp. 159-174

Résumé

The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let $P(n)$ stand for the largest prime factor of $n$. Then define $f(x,p):=\#\{ n\le x\mid P(n)=p\} $. If $f(x,p)$ is considered as a function of $p$, for $2\le p\le x$, the primes in the interval $[2,x]$ belong to three intervals $I_1(x)=[2,v(x)]$, $I_2(x)=\mathopen ]v(x),w(x)\mathclose [$ and $I_3(x)=[w(x),x]$, with $v(x) w(x)$, such that $f(x,p)$ increases for $p\in I_1(x)$, reaches its maximum value in $I_2(x)$, in which interval it oscillates, and finally decreases for $p\in I_3(x)$. In fact, we show that $v(x)\ge \sqrt{\mathop {\rm log}\nolimits x}$ and $w(x)\le \sqrt{x}$. We also provide several conditions on primes $p\le q$ so that $f(x,p)\ge f(x,q)$.

Détail

DOI : 10.4064/cm88-2-1

Keywords: main objective paper analyze unimodal character frequency function largest prime factor stand largest prime factor define mid considered function primes interval belong three intervals mathopen mathclose increases reaches its maximum value which interval oscillates finally decreases sqrt mathop log nolimits sqrt provide several conditions primes