素数间隙中粗数的重数常数
Multiplicity Constants for Rough Numbers in Prime Gaps
- Yeditepe University（耶迪特佩大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文研究素数间隙中粗数的重数常数，定义$N_r(X)$为满足$X\le p_n\le2X$且粗数数量$M_n<r$的$n$的个数，通过局部筛模型结合矩界等方法得到$c_r$的渐近结果，还给出$c\ge2.7413$的下界。
AI中文摘要：
设$p_n$为第$n$个素数，$g_n=p_{n+1}-p_n$。遵循Gafni和Tao的定义，若整数$m$满足$p_n<m<p_{n+1}$且其最小素因子至少为$g_n$，则称$m$为该间隙的粗数；记$M_n$为这类整数的数量。Gafni与Tao证明，满足$X\le p_n\le 2X$且$M_n=0$的$n$的数量为$O(X/\log^2 X)$，且在迪克森-哈代-李特尔伍德素元组猜想的某种形式下，其渐近于$cX/\log^2 X$。我们研究$N_r(X)$，即满足$X\le p_n\le 2X$且$M_n<r$的$n$的数量。对每个$r\ge1$，我们通过端点条件局部筛模型定义常数$c_r$，其中$c_1=c$；并无条件证明，当$r\ge3$时，$c_r=h_r^*+O_\varepsilon(r^{1/2}(\log r)^{1+\varepsilon})$，其中$h_r^*$是满足$h\prod_{p<h}(1-1/p)\ge r$的最小偶数$h\ge2$。由此可得，对某个绝对常数$\kappa>0$，$c_r=H_r(1+O(e^{-\kappa\sqrt{\log r}}))$，其中$H_r>e$满足$e^{-\gamma}H_r/\log H_r=r$（$\gamma$为欧拉常数），且$c_r=e^\gamma r(\log r+\log\log r+\gamma+(\log\log r+\gamma)/\log r+O((\log\log r)^2/(\log r)^2))$。证明结合了短区间内简化剩余的Montgomery-Vaughan矩界与端点条件的精确移除。对每个固定$r$，Gafni和Tao的论证可扩展为：无条件下$N_r(X)\ll_r X/\log^2 X$，在其猜想下$N_r(X)\sim c_rX/\log^2 X$。局部模型的精确枚举给出$c\ge2.7413$。
英文摘要：
Let $p_n$ be the $n$th prime and $g_n=p_{n+1}-p_n$. Following Gafni and Tao, an integer $m$ with $p_n<m<p_{n+1}$ is rough for the gap if its least prime factor is at least $g_n$; let $M_n$ be the number of such integers. Gafni and Tao proved that the number of $n$ with $X\le p_n\le 2X$ and $M_n=0$ is $O(X/\log^2 X)$, and that it is asymptotic to $cX/\log^2 X$ under a form of the Dickson-Hardy-Littlewood prime tuples conjecture. We study $N_r(X)$, the number of $n$ with $X\le p_n\le 2X$ and $M_n<r$. For every $r\ge 1$ we define a constant $c_r$ by an endpoint-conditioned local sieve model, with $c_1=c$, and we prove unconditionally that $c_r=h_r^*+O_\varepsilon(r^{1/2}(\log r)^{1+\varepsilon})$ for $r\ge 3$, where $h_r^*$ is the least even $h\ge 2$ with $h\prod_{p<h}(1-1/p)\ge r$. Consequently $c_r=H_r(1+O(e^{-κ\sqrt{\log r}}))$ for an absolute constant $κ>0$, where $H_r>e$ solves $e^{-γ}H_r/\log H_r=r$ ($γ$ is Euler's constant), and $c_r=e^γr(\log r+\log\log r+γ+(\log\log r+γ)/\log r+O((\log\log r)^2/(\log r)^2))$. The proof combines the Montgomery-Vaughan moment bounds for reduced residues in short intervals with an exact removal of the endpoint conditioning. For each fixed $r$, the argument of Gafni and Tao extends to give $N_r(X)\ll_r X/\log^2 X$ unconditionally, and $N_r(X)\sim c_rX/\log^2 X$ under their conjecture. Exact enumeration of the local model gives $c\ge 2.7413$.
