## On exponential sums over primes and application in Waring-Goldbach problem.(English)Zbl 1100.11025

Author’s abstract: We prove the following estimate on exponential sums over primes: Let $$k \geq 1$$, $$\beta_k = 1/2 + \log k / \log 2$$, $$x \geq 2$$ and $$\alpha = a/q + \lambda$$ subject to $$(a,q)=1$$, $$1 \leq a \leq q$$ and $$\lambda \in \mathbb{R}$$ . Then $\sum_{x < m \leq 2x} \Lambda(m) e(\alpha m^k) ~\ll~ (d(q))^{\beta_k} (\log x)^c \left( x^{1/2} \sqrt{q (1+ | \lambda | x^k)} + x^{4/5} + \frac{x}{\sqrt{q(1+| \lambda| x^k)}} \right).$ As an application, we prove that with at most $$O(n^{7/8 + \varepsilon})$$ 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.

### MSC:

 11L20 Sums over primes 11P05 Waring’s problem and variants 11P32 Goldbach-type theorems; other additive questions involving primes