短区间内最小素因子的和
On the sum of least prime factors in short intervals
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中文总结 AI 辅助
本文研究短区间内合数的最小素因子之和，估计了均值渐近常数，得出区间和的均值与二阶矩性质，在弱假设下肯定回答Erdős–Graham问题，无条件一致陈述仍待解决。
中文摘要 AI 辅助
设$p(n)$表示$n$的最小素因子，$L_C(x)=Cx^{1/2}(\boldsymbol{\text{log}} x)^2$（$C>0$）。本文研究了Erdős与Graham提出的问题：短区间$[x, x+L_C(x)]$内合数$n$对应的$p(n)/n$之和。（i）估计了均值渐近中的常数$c=8$，即$S(x)=\boldsymbol{\text{sum}}_{n<x,\text{ }n\text{ 为合数}}\frac{p(n)}{n}=\frac{c x^{1/2}}{(\boldsymbol{\text{log}} x)^2}\bigl(1+O\bigl(\frac{1}{\boldsymbol{\text{log}} x}\bigr)\bigr)+O\bigl(\frac{x^{1/3}}{\boldsymbol{\text{log}} x}\bigr)$；（ii）对任意固定$C>0$，合数上的区间和$\boldsymbol{\text{mu}}_C(x):=\boldsymbol{\text{sum}}_{x\boldsymbol{\text{≤}}n\boldsymbol{\text{≤}}x+L_C(x)}p(n)/n$的均值为$4C$，即$\frac{1}{X}\boldsymbol{\text{sum}}_{x\boldsymbol{\text{≤}}X}\boldsymbol{\text{mu}}_C(x)=4C+O_C(1/\boldsymbol{\text{log}} X)$，二阶矩为$\frac{1}{X}\boldsymbol{\text{sum}}_{x\boldsymbol{\text{≤}}X}(\boldsymbol{\text{mu}}_C(x)-4C)^2=O_C((\boldsymbol{\text{log}} X)^{-2})$，特别地，几乎所有$x$满足$\boldsymbol{\text{mu}}_C(x)=4C+o(1)$；（iii）在关于长度为$(\boldsymbol{\text{log}} y)^{2+o(1)}$区间内素数的弱Cramér型假设下，$\boldsymbol{\text{mu}}_C(x)=4C+O_C(1/\boldsymbol{\text{log}} x)$对所有$x$一致成立，给出了Erdős–Graham问题的肯定回答；无条件下的一致陈述仍未解决，证明该陈述需解决尺度为$(\boldsymbol{\text{log}} y)^2$的短区间素数估计问题。
英文摘要
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 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 $μ_C(x):=\sum_{x\le n\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\frac{1}{X}\sum_{x\le X}μ_C(x)=4C+O_C(1/\log X)$, and second moment $\frac{1}{X}\sum_{x\le X}(μ_C(x)-4C)^2=O_C((\log X)^{-2})$. In particular $μ_C(x)=4C+o(1)$ for almost all $x$. (iii)Under a weak Cramér-type hypothesis on primes in intervals of length $(\log y)^{2+o(1)}$, the estimate $μ_C(x)=4C+O_C(1/\log x)$ holds uniformly in $x$, giving an affirmative answer to the Erdős--Graham question. Unconditionally, the uniform statement remains open; proving the uniform statement unconditionally would require resolving short-interval prime estimates at scale $(\log y)^2$.
