Estimates of complete rational trigonometric sums and sums of Dirichlet characters (Q1425510)

The authors give a.o. a proof of the estimate of \textit{L.-K. Hua} [``Method of trigonometric sums and its application in number theory'', Nauka, Moscow (1964; Zbl 0122.05202)] of the complete rational trigonometric sum \[ S(f,Q)\leq c_nQ^{1-1/n}, \quad f\in F_n(Q), \] where \(F_n(Q)\) is the set of all polynomials \(f(x) = a_nx^n+...+a_1x+a_0 \in \mathbb Z[x]\) with the condition \((a_n,...,a_1,Q)=1\) and \[ S(f,Q) = \sum\limits_{x=1}^Q \exp\bigg( 2\pi i\, \frac{f(x)}{Q}\bigg). \]