Sums of four squares of primes

TL;DR

This paper investigates the distribution of integers congruent to 4 mod 24 that cannot be expressed as the sum of four prime squares, providing an improved upper bound on their count.

Contribution

The authors improve the upper bound on the number of such integers from an exponent of 7/20 to 11/32, advancing understanding of sums of prime squares.

Findings

Established that E(N)  N^{11/32}

Improved previous bound from N^{7/20}

Enhanced methods for analyzing sums of four prime squares

Abstract

Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4 \pmod{24}$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.