Prime Solutions of Diagonal Diophantine Systems

Abstract

An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the Vinogradov Mean Value Theorem this yields an asymptotic formula for solutions of Vinogradov systems and in conjunction with Hooley's work on seven cubes this yields a conditional result for the Waring-Goldbach problem on seven cubes of primes, contingent on Hooley's form of the Riemann hypothesis.