On the sum of least prime factors in short intervals

Abstract

Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=C{x}^{1/2}(\log x{)}^2$ 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 Erdős 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 $\frac{1}{X}{\sum }_{x\le X}({\mu }_C(x)-4C{)}^2=O_C(1)$. (iii)Under the Maier-consistent bounded-fluctuation hypothesis, ${\mu }_C(x)=4C+O_C(1)$ holds uniformly in $x$, and the Erdős--Graham question has an affirmative answer. Unconditionally, the uniform statement remains open; proving it would require resolving short-interval prime estimates at scale $(\log y{)}^2$.