Exponential sums over primes and the prime twin problem

摘要

The purpose of this paper is to obtain an effective estimate of the exponential sum (where e(α)=e~(2πiα), α,β∈?, (a,q)=1 and Λ is the von Mangoldt function) in the range. It improves Daboussi's estimate [2, Theorem 1] in the range q≤(log x)~D and x(log x)~(-D)≤q, D>0 and is valid in a wider range for β.