On exponential sums over primes and application in Waring-Goldbach problem
On exponential sums over primes and application in Waring-Goldbach problem
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DOI：
10.1360/03ys0341
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发表时间：
2005
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影响因子：
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通讯作者：
Xiumin Ren
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文献类型：
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作者：
Xiumin Ren
摘要在本文中，我们证明了以下对素数指数和的估计：设 k ⩾ 1、βk = 1/2 + log k/log 2、x ⩾ 2 和 α = a/q + λ ，且满足 (a,q) = 1、1 ⩽ a ⩽ q 和 λ ε ℝ。然后 $$\sum\limits_{x < m \leqslant 2x} {\Lambda (m)e(\alpha m^k ) \ll (d(q))^{\beta _k } (\log x)} ^c \left( {x^{1/2} \sqrt {q(1 + \left| \lambda \right|x^k )} + x^{4/5} + \frac{x}{{\sqrt {q(1 + \left| \lambda \right|x^k )} }}} \right).$$ 作为一个应用，我们证明除了 O(N7/8+ε) 次例外，所有满足某些必要同余条件的 N 以内的正整数都是三个素数的平方和。这个结果与之前在广义黎曼假设下建立的结果一样有力。
AbstractIn this paper, we prove the following estimate on exponential sums over primes: Let k ⩾ 1, βk = 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(N7/8+ε) 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.