Sum of short exponential sums with prime numbers

Abstract

For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K\le y<x$, where $f(u)=\alpha u^n+{\alpha }_{n-1}{u}^{n-1}+\dots +{\alpha }_{1}u$ 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 $\mathcal{L}=\ln x$, an estimate of the form $$\sum _{k=1}^K\left|\sum _{x-y<p\le x}e(kf(p))\right|\ll Ky{\left(\frac{1}{q}+\frac{1}{y}+\frac{q}{K{y}^n}+\frac{1}{K^{2^{n-1}}}\right)}^{2^{-n-1}}{\mathcal{L}}^{\frac{n^2}{2^{n+1}}},$$ 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.