Distribution Of Integers That Are Sums Of A Prime And Two Squares Of Primes

In this paper we prove that with at most O(N5/14+ε) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of a prime and two squares of primes. This improves substantially the previous results in this direction.Usually we just use the circle method to deal with many additive prime problems, including the one in this paper. There is always a vital obstacle in our research, that is, the estimates of exponential sums in minor arcs. If we only use the circle method, we had better use the estimates of A. Ghosh and Xiumin Ren, but this can do the limited improvement to our problem.Here we will combine the circle method with the sieve method, developed by G. Harman and A. V. Kumchev. By introducing p, a non-trivial lower bound for the characteristic function of the set of primes, with many necessary properties, we can improve the estimate in minor arcs so that the better result will be obtained.