On sums of squares of primes II

TL;DR

This paper advances the understanding of integers that cannot be expressed as sums of three or four prime squares by refining exponential sum estimates and sieve techniques, improving bounds on exceptional sets.

Contribution

It corrects a previous oversight and introduces enhanced exponential sum estimates and sieve methods to better bound the size of exceptional sets.

Findings

Improved bounds for the maximal size of exceptional sets.

Refined exponential sum estimates for sums of prime squares.

Correction of a key oversight from the previous work.

Abstract

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.