Weyl sum

A trigonometric sum of the form

$$S(f)=\sum_{1\leq x\leq P}e^{2\pi if(x)},\label{*}\tag{*}$$

where

$$f(x)=\alpha_nx^n+\dotsb+\alpha_1x$$

and $\alpha_n,\dots,\alpha_1$ are arbitrary real numbers. Weyl sums are used in solving many familiar problems number theory. The first method for obtaining non-trivial estimates of the sums \eqref{*} was developed in 1916 by H. Weyl (cf. Weyl method). Essentially better estimates of Weyl sums were obtained in 1934 by I.M. Vinogradov, who used his own new method for estimating trigonometric sums (cf. Vinogradov method).