Sums of Three Prime Squares

Mikawa, Hiroshi ; Peneva, Temenoujka

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 549-558

Résumé

Let $A, \epsilon > 0$ be arbitrary. Suppose that $x$ is a sufficiently large positive number. We prove that the number of integers $n \in (x, x+x^\theta]$, satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is $\ll x^\theta(\log x)^{-A}$, provided that $\frac{7}{16} + \epsilon \leq \theta \leq 1$.