On exponential sums over primes and application in Waring-Goldbach problem

Abstract

In this paper, we prove the following estimate on exponential sums over primes: Let k ⩾ 1, β k = 1/2 + log k/log 2, x ⩾ 2 and α = a/q + λ subject to (a,q) = 1, 1 ⩽ a ⩽ q, and λ ∈ ℝ. Then $$\sum\limits_{x < m \leqslant 2x} {\Lambda (m)e(\alpha m^k ) \ll (d(q))^{\beta _k } (\log x)} ^c \left( {x^{1/2} \sqrt {q(1 + \left| \lambda \right|x^k )} + x^{4/5} + \frac{x}{{\sqrt {q(1 + \left| \lambda \right|x^k )} }}} \right).$$ As an application, we prove that with at most O(N 7/8+e) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.