An asymptotic formula with power-saving error term for counting prime solutions to a binary additive problem

Abstract

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box $[-X,X{]}^4$ that satisfy the determinant equation $x_1{x}_2-x_3{x}_4=r$ for $r\ne 0$ with two of entries to be prime. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes $p$.

15 pages, Submitted