Sum of short exponential sums with prime numbers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915543626285056 |
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| author | Rakhmonov, Firuz |
| author_facet | Rakhmonov, Firuz |
| contents | For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = αu^n + α_{n-1}u^{n-1} + \ldots + α_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $α$ is a real number such that $\left|α- \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09102 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum of short exponential sums with prime numbers Rakhmonov, Firuz Number Theory For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = αu^n + α_{n-1}u^{n-1} + \ldots + α_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer, $α$ is a real number such that $\left|α- \frac{a}{q}\right| \le \frac{1}{q^2}$, $(a, q) = 1$, $q \ge 1$ 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. |
| title | Sum of short exponential sums with prime numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.09102 |