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On exponential sums over primes in short intervals

Authors:	Guangshi Lü Huixue Lao

Institution:	(1) Shandong University, Jinan, China;(2) Shandong Normal University, Jinan, China

Abstract:	Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals $S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha})$ for all α ∈ 0,1] whenever $x^{\frac{2}{3}+{\varepsilon}}\le y \le x$ . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

Keywords:	2000 Mathematics Subject Classification: 11P32 11P05 11N36 11P55
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