Multiplicity Constants for Rough Numbers in Prime Gaps

Abstract

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 $c_r=H_r(1+O(e^{-\kappa\sqrt{\log r}}))$ for an absolute constant $\kappa>0$, where $H_r>e$ solves $e^{-\gamma}H_r/\log H_r=r$ ($\gamma$ is Euler's constant), and $c_r=e^\gamma r(\log r+\log\log r+\gamma+(\log\log r+\gamma)/\log r+O((\log\log r)^2/(\log r)^2))$. 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 $c\ge 2.7413$.