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

TL;DR

This paper derives an asymptotic formula with a power-saving error term for counting integer solutions within a growing box that satisfy a determinant equation with two prime entries, using advanced harmonic analysis techniques.

Contribution

It introduces a new asymptotic counting method for solutions to a binary additive problem involving primes and determinant conditions, with explicit error bounds.

Findings

Established an asymptotic formula with power-saving error for solutions involving primes.

Applied Poisson summation and Kloosterman sum estimates to prime-related counting.

Enhanced understanding of prime solutions in determinant equations within expanding regions.

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 $x_1x_2-x_3x_4=r$ for $r \neq 0$ 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 $p$.