A Bombieri-type Theorem for Exponential Sums

Let fk(n) be the characteristic function of n with Ω(n) = k,and Tk(x,α) =∑n≤xfk(n)e(nα).The main purpose of this paper is to establish a Bombieri-type mean-value theorem for Tk(x,α),via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions.The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method.