On the representation of large integers as sums of four almost equal squares of primes

Abstract

In this paper, we are able to establish two localized results on the conjecture that each large integer congruent to 4 modulo 24 can be written as the sum of four squares of primes. The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method.