On certain exponential sums over primes

Let f(x)f(x) be a real valued polynomial in x   of degree k⩾4k⩾4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity|∑p⩽N(logp)e(f(p))| whenever the leading coefficient α   of f(x)f(x) is of type 1.