On the sum of least prime factors in short intervals

Abstract

Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\,x+L_C(x)]$, a question raised by Erd\H{o}s and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \[ S(x)=\sum_{{n<x,\ n\ \text{composite}}}\frac{p(n)}{n}=\frac{c\,x^{1/2}}{(\log x)^2}\Bigl(1+O\Bigl(\frac1{\log x}\Bigr)\Bigr) +O\Bigl(\frac{x^{1/3}}{\log x}\Bigr). \] (ii)For every fixed $C>0$, the window sums $\mu_C(x):=\sum_{x\le n\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\frac{1}{X}\sum_{x\le X}\mu_C(x)=4C+O_C(1/\log X)$, and second moment $\frac{1}{X}\sum_{x\le X}(\mu_C(x)-4C)^2=O_C((\log X)^{-2})$. In particular $\mu_C(x)=4C+o(1)$ for almost all $x$. (iii)Under a weak Cram\'{e}r-type hypothesis on primes in intervals of length $(\log y)^{2+o(1)}$, the estimate $\mu_C(x)=4C+O_C(1/\log x)$ holds uniformly in $x$, giving an affirmative answer to the Erd\H{o}s--Graham question. Unconditionally, the uniform statement remains open; proving the uniform statement unconditionally would require resolving short-interval prime estimates at scale $(\log y)^2$.