Estimation of exponential sums over primes in short intervals I

Let Lambda(n) and mu(n) be the von Mangoldt function and Mobius function, r espectively, x real and y "small" compared with x. This paper gives, for th e first time, a non-trivial estimate of the sum S-2(x, y; alpha)= Sigma(x<n less than or equal to x+y) Lambda(n)e(n(2)alpha ) for all alpha is an element of [0, 1] whenever x(11/16+epsilon)less than or equal to y less than or equal to x.Correspondingly, it is also proved that Sigma(x less than or equal to n less than or equal to x+y) mu(n)e(n(2)alpha ) much less than y(log x)(-A) uniformly for all real alpha and any A>0, and in the same range of y.