A Bombieri-type theorem for exponential sums (Q381074)

Let \(\Omega(n)\) be the number of prime divisors of \(n>1\), and let \(f_k(n)\) denote the characteristic function of \(n\) with \(\Omega(n)=k\). This paper establishes a Bombieri-type mean value theorem for the exponential sum \[ T_k(x,\alpha)=\sum_{n\leq x}f_k(n)e(n\alpha), \] by using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet \(L\)-functions.