Prime Solutions of Diagonal Diophantine Systems

TL;DR

This work advances the study of prime solutions to general diagonal Diophantine systems by developing a conditional circle-method framework that ties the existence of mean-value bounds to asymptotic representations by primes. It introduces a generating-function setup, a precise major/minor-arc decomposition, and sharp oscillatory integral bounds to extract a singular-series/singular-integral structure, yielding an asymptotic formula $R(P) \sim C P^{s-K}$ under locality and mean-value hypotheses. In key special cases, notably Vinogradov systems, it shows that with sufficient mean-value power, $R(P)$ matches the exponent predicted by the corresponding integer-problem theory, and conditional on Hooley’s HW-type hypothesis, it implies a Waring-Goldbach result for seven cubes of primes. The analysis links recent Vinogradov mean-value bounds to prime-diophantine representations, providing a flexible, conditional toolkit for future unconditional progress as mean-value bounds improve. Overall, the paper extends the circle-method frontier for prime solutions in Diophantine systems and clarifies how local solvability and analytic bounds govern the asymptotic count.

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.

Theorems & Definitions (48)

• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• Lemma 3.1