Published online by Cambridge University Press: 29 November 2017
Let $f$ be a polynomial of degree
$k>1$ with irrational leading coefficient. We obtain results of the form
$$\begin{eqnarray}\Vert f(p)\Vert <p^{-\unicode[STIX]{x1D70E}}\end{eqnarray}$$
$p$ that supersede those of Harman [Trigonometric sums over primes I. Mathematika28 (1981), 249–254; Trigonometric sums over primes II. Glasg. Math. J.24 (1983), 23–37] and Wong [On the distribution of
$\unicode[STIX]{x1D6FC}p^{k}$ modulo 1. Glasg. Math. J.39 (1997), 121–130].