Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums

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 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. Finally, these estimates are applied to evaluate mixed fourth moments of Dirichlet character sums over integers and primes, yielding non-trivial bounds. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes.

22 pages, Submitted