关于孪系函数在素数参数下二元谐波系数相关性的研究
数论
2025-11-14 v1
摘要
设 { λ f ( n ) } n ≥ 1 \{\lambda_f(n)\}_{n \geq 1} { λ f ( n ) } n ≥ 1 f f f L ( s , f , χ ) L(s, f, \chi) L ( s , f , χ ) χ \chi χ q q q X 2 / 3 + ε ≪ H ≪ X 1 − ε X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon} X 2/3 + ε ≪ H ≪ X 1 − ε ε > 0 \varepsilon>0 ε > 0 A ≥ 1 A\geq 1 A ≥ 1 ∑ 1 ≤ ∣ h ∣ ≤ H ∣ ∑ X < n , m ≤ 2 X n − m = h λ f ( n ) Λ ( n ) λ f ( m ) Λ ( m ) ∣ 2 ≪ A H X 2 ( log X ) A \sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} \lambda_f(n)\Lambda(n)\lambda_f(m)\Lambda(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}} 1 ≤ ∣ h ∣ ≤ H ∑ X < n , m ≤ 2 X n − m = h ∑ λ f ( n ) Λ ( n ) λ f ( m ) Λ ( m ) 2 ≪ A ( log X ) A H X 2 L ( s , f ) L(s, f) L ( s , f ) X 1 / 3 + ε ≪ H ≪ X 1 − ε X^{1/3+\varepsilon}\ll H \ll X^{1-\varepsilon} X 1/3 + ε ≪ H ≪ X 1 − ε f f f X ε ≪ H ≪ X 1 − ε X^{\varepsilon}\ll H\ll X^{1-\varepsilon} X ε ≪ H ≪ X 1 − ε A ≥ 1 A\geq 1 A ≥ 1 ∑ f ∈ H k ω f ∑ 1 ≤ ∣ h ∣ ≤ H ∣ ∑ X < n , m ≤ 2 X n − m = h λ f ( n ) Λ ( n ) λ f ( m ) Λ ( m ) ∣ 2 ≪ A H X 2 ( log X ) A , \sum_{f\in \mathcal{H}_k}\omega_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} \lambda_f(n)\Lambda(n)\lambda_f(m)\Lambda(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}}, f ∈ H k ∑ ω f 1 ≤ ∣ h ∣ ≤ H ∑ X < n , m ≤ 2 X n − m = h ∑ λ f ( n ) Λ ( n ) λ f ( m ) Λ ( m ) 2 ≪ A ( log X ) A H X 2 , H k \mathcal{H}_k H k k k k S L ( 2 , Z ) \mathrm{SL}(2, \mathbb{Z}) SL ( 2 , Z ) ω f \omega_f ω f f ∈ H k f\in \mathcal{H}_k f ∈ H k
引用
@article{arxiv.2511.10594,
title = {On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments},
author = {Jiseong Kim and Kunjakanan Nath},
journal= {arXiv preprint arXiv:2511.10594},
year = {2025}
}
备注
22 pages, comments are welcome