On exponential sums over primes in short intervals

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.