The quadratic Waring-Goldbach problem
The purpose of this paper is to show a new bound for the number \(E(N)\), say, of the natural numbers \(n\) such that \(n\leq N\), \(n\equiv4\pmod {24}\), and \(n\) cannot be written as the sum of four squares of primes. Actually, the authors establish the bound \(E(N)\ll N^{3/8+\varepsilon}\) with any fixed \(\varepsilon>0\). The proof is based on the circle method, and also on a theorem concerning the major arc integrals due to [\textit{J. Liu}, Q. J. Math. 54, No. 4, 453--462 (2003; Zbl 1080.11071)]. The minor arc integrals are handled by the method of [\textit{T. D. Wooley}, Proc. Lond. Math. Soc. (3) 85, 1--21 (2002; Zbl 1039.11066)]. In order to obtain the strong bound for \(E(N)\) in the theorem, however, the latter minor arcs have to be narrower than those corresponding to the major arcs in the above theorem of Liu. This remaining part is managed by a certain estimate for a quadratic Weyl sum over primes on the intermediate arcs, that is, the gap between the major and minor arcs. The latter estimate is proved by two different ways, and indeed most part of the paper is devoted to these proofs. The first proof appeals to zero density estimates for Dirichlet \(L\)-functions, and in the second one, the argument is reduced to the estimate for Dirichlet polynomials shown in the above work of Liu, via application of Heath-Brown's identity for the von Mangoldt function.
scientific article; zbMATH DE number 2091713
scientific article