Exponential sum bound of Mordell and Hua

We improve estimates for the exponential sum S(f,q)=& sum;(q )(x=1)e(q)(f(x)), where e(q)(& sdot;) = e(2 pi i & sdot;/q) and f(x) is a primitive polynomial over Z. Let R(f,q)=|S(f,q)|/q1-1/k, with k the degree of f, and R(k,q) be the maximum value of R(f,q) over primitive polynomials of degree k. Among other results, we show that for any prime power pm with 5 <= p <= 2k-1 we have R(k,p(m)) <= p(2/p+1+1/p). In particular, R(k,p(m)) <= 2.815 for any k and prime power pm. We also show that for any positive integer q, R(k,q) <= e(k+1/2 pi log2k + 6logk-4.88891) for k < 4.62 & sdot;10(12) unconditionally, and for all k >= 1 on the assumption of the Riemann Hypothesis. Refined estimates are given for small k.