A Bombieri-type theorem for exponential sums

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Let fk(n) be the characteristic function of n with Ω(n) = k, and $$T_k \left( {x,\alpha } \right) = \sum\limits_{n \leqslant x} {f_k \left( n \right)e\left( {n\alpha } \right)} .$$ 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.