On the unimodal character of the frequency function of the largest prime factor
Volume 88 / 2001
Abstract
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 $f(x,p)\ge f(x,q)$.
