Abstract:
For sufficiently large integers $K$, $x$, $y$, $q$, subject to $K\le y<x$, $n$ — fixed natural number, $\alpha$ — real, $\left|\alpha-\frac{a}{q}\right|\le\frac1{q^2}$, $(a,q)=1$, $q\ge1$, an estimate of the form $$ \sum_{k=1}^K\left|\sum_{x-y<p\le x}e(\alpha kp^n)\right| \ll Ky\left(\frac1q+\frac1y+\frac{q}{Ky^n}+\frac1{K^{2^{n-1}}}\right)^{2^{-n-1}}\mathscr{L}^{\frac{n^2}{2^{n+1}}}, $$ which is a strengthening and generalization of I. M. Vinogradov's theorem on the distribution of fractional parts of $\{\alpha p\}$.
Keywords:Short exponential sum of G. Weyl with prime numbers, uniform distribution modulo unity, non-trivial estimate, fractional part.
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