Correlations of Almost Primes

Natalie Evans

Published 2021-02-24Version 1

Let $p_1,\ldots,p_5$ be primes with $p_1p_2,p_5\in(X,2X]$, let $h$ be a non-zero integer and $A > 3$. Supposing that $p_1,p_3$ lie in a certain range depending on $X$, we prove an asymptotic for the weighted number of solutions to $p_3p_4=p_1p_2+h$ which holds for almost all $0 < |h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X\log^{-A}X$. Using the same methods and supposing instead that $p_1p_2, p_3p_4$ have the typical factorisation we prove an asymptotic which holds for almost all $0 < |h|\leq H$ with $\exp\left((\log X)^{1-o(1)}\right)\leq H\leq X\log^{-A}X$ and an asymptotic for the weighted number of solutions to $p_3p_4=p_5+h$ which holds for almost all $0 < |h|\leq H$ with $X^{1/6+\varepsilon}\leq H\leq X\log^{-B}X$ and $B > 5$.