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

Ren, Xiumin

Sci. China, Ser. A 48, No. 6, 785-797 (2005).

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.

Reviewer: Jürgen Hinz (Marburg)

MSC:

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