Exponential sums over primes and the prime twin problem

The purpose of this paper is to obtain an effective estimate of the exponential sum $sum_{nle x}Lambda(n)eleft(left(frac{a}{q}+betaright)nright)$ (where e(??)=e ^{2?? i ??} , ??,?????, (a,q)=1 and ?? is the von Mangoldt function) in the range ${(log x)}^{1/2+varepsilon}le qle frac{x}{{(logloglog x)}^{1+varepsilon}}$ and $|beta| . 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 ??.