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.


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