素数短指数和
数论
2025-10-13 v1
摘要
对于满足 K ≤ y < x K \le y < x K ≤ y < x K K K x x x y y y q q q f ( u ) = α u n + α n − 1 u n − 1 + … + α 1 u f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u f ( u ) = α u n + α n − 1 u n − 1 + … + α 1 u n n n n n n α \alpha α ∣ α − a q ∣ ≤ 1 q 2 \left|\alpha - \frac{a}{q}\right| \le \frac{1}{q^2} α − q a ≤ q 2 1 ( a , q ) = 1 (a, q) = 1 ( a , q ) = 1 q ≥ 1 q \ge 1 q ≥ 1 L = ln x \mathscr{L} = \ln x L = ln x ∑ k = 1 K ∣ ∑ x − y < p ≤ x e ( k f ( p ) ) ∣ ≪ K y ( 1 q + 1 y + q K y n + 1 K 2 n − 1 ) 2 − n − 1 L n 2 2 n + 1 , \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}}}, k = 1 ∑ K x − y < p ≤ x ∑ e ( k f ( p )) ≪ K y ( q 1 + y 1 + K y n q + K 2 n − 1 1 ) 2 − n − 1 L 2 n + 1 n 2 ,
引用
@article{arxiv.2510.09102,
title = {Sum of short exponential sums with prime numbers},
author = {Firuz Rakhmonov},
journal= {arXiv preprint arXiv:2510.09102},
year = {2025}
}