Sum of short exponential sums with prime numbers

Abstract

For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $\alpha$ is a real number such that $\left|\alpha - \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ and $\mathscr{L} = \ln x$, an estimate of the form $$ \sum_{k=1}^K \left| \sum_{x - y < p \le x} e(kf(p)) \right| \ll K y \left( \frac{1}{q} + \frac{1}{y} + \frac{q}{K y^n} + \frac{1}{K^{2^{n-1}}} \right)^{2^{-n-1}} {\mathscr{L}}^{\frac{n^2}{2^{n+1}}}, $$ is obtained, which represents a strengthening and generalization of the corresponding estimate of I.M.Vinogradov. Keywords: short exponential sum of G.Weyl with prime numbers, uniform distribution modulo one, nontrivial estimate, fractional part. Bibliography: 18 references.