Estimate for exponential sums and its applications

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established.