An L-function free proof of Hua's Theorem on sums of five prime squares

TL;DR

This paper offers a new proof of Hua's theorem on representing large integers congruent to 5 mod 24 as sums of five prime squares, avoiding L-function techniques by using a sieve and transference principle.

Contribution

It introduces an L-function free proof of Hua's theorem utilizing the transference principle and sieve methods, simplifying the original approach.

Findings

Proof avoids L-function zero distribution results

Establishes pseudorandomness of the prime majorant

Accurately evaluates Gaussian sums and Jacobi symbols

Abstract

We provide a new proof of Hua's result that every sufficiently large integer N\equiv 5(mod\,24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of $L$-functions. Here, we present a proof based on the transference principle first introduced by Green. Using a sieve theoretic approach similar to the work by Shao, we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudorandomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.