The exceptional set in the four prime squares problem
A theorem of \textit{L. K. Hua} [Q. J. Math., Oxf. Ser. 9, 68-80 (1938; Zbl 0018.29404)] states that each large integer congruent to \(5\pmod {24}\) can be written as a sum of five squares of primes. In view of this result and Lagrange's theorem of four squares, it is conjectured that each large integer \(n\equiv 4\pmod {24}\) is a sum of four squares of primes. This paper establishes the following approximation to this conjecture. Let \(N\geq 2\), and \(E(N)\) the number of all the positive integers \(n\equiv 4\pmod {24}\) not exceeding \(N\) which cannot be written as four squares of primes. Then \(E(N)\ll N^{13/15+ \varepsilon}\). The first result in this direction is due to the result of L. K. Hua cited above, who proved that \(E(N)\ll N\log^{-4} N\) for some absolute constant \(A>0\). Later \textit{W. Schwarz} [J. Reine Angew. Math. 206, 78-112 (1961; Zbl 0102.28201)] showed that the above \(A\) can be taken arbitrarily. In this paper the authors apply the circle method together with the large sieve inequality, \textit{P. X. Gallagher}'s lemma [Invent. Math. 11, 329-339 (1970; Zbl 0219.10045)] and \textit{D. R. Heath-Brown}'s identity [Can. J. Math. 34, 1365-1377 (1982; Zbl 0494.10027)], and are able to avoid the influence of the Deuring-Heilbronn phenomenon.
scientific article; zbMATH DE number 1448992