【摘 要】
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In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q)
【机 构】
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Department of Mathematics
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In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣx＜m≤2x∧(m)e(αmk)＜＜(d(q))βk(logx)c(x1/2√q(1+|λ|xk)+x4/5+x/√q(1+|λ|xk)).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.
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