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(1.1) n = p1 + p 2 2 + p 2 3. To be more precise, let H = {n ≥ 1 : n ≡ 3 (mod 24), n 6≡ 0 (mod 5)}, and let E(N) be the number of n ∈ H not exceeding N that cannot be represented as (1.1). Then Hua’s result actually states that E(N) N log−AN for some positive constant A. Later Schwarz [16] proved that Hua’s estimate holds for arbitrary A > 0. And in 1993, Leung and Liu [10] improved the upper bound of E(N) to N1−δ, where δ > 0 is some computable absolute constant depending on, among other things, the constants in the Deuring– Heilbronn phenomenon. Recently, Bauer, Liu, and Zhan [1] have dealt with this problem via a different approach without the Deuring–Heilbronn phenomenon, and established E(N) N 77/80+ε, where ε > 0 is arbitrary. In this paper, we make the following improvement.

Distribution of integers that are sums of three squares of primes