具有超临界除数扰动的随机乘性函数的部分和
数论
2026-04-08 v1 泛函分析
概率论
摘要
设 f f f α ∈ R \alpha\in \mathbb{R} α ∈ R d α d_\alpha d α α \alpha α α ∈ ( 1 , 2 ) \alpha \in (1,2) α ∈ ( 1 , 2 ) E { ∣ 1 x ∑ n ≤ x d α ( n ) f ( n ) ∣ 2 q } ≪ ( log x ) 2 q ( α − 1 ) ( log log x ) 3 α q / 2 ( 1 − α q ) + 1 \mathbb{E}\bigg\{\Big|\frac{1}{\sqrt{x}}\sum_{n\leq x} d_\alpha(n)f(n)\Big|^{2q}\bigg\} \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}(1-\alpha q)+1} E { x 1 n ≤ x ∑ d α ( n ) f ( n ) 2 q } ≪ ( log log x ) 3 α q /2 ( 1 − α q ) + 1 ( log x ) 2 q ( α − 1 ) q ∈ [ 0 , 1 / α ] q\in [0,1/\alpha] q ∈ [ 0 , 1/ α ] x x x α = 1 \alpha=1 α = 1 f f f α = 1 \alpha=1 α = 1 q = 1 / 2 q=1/2 q = 1/2 lim T → ∞ 1 T ∫ T 2 T ∣ ∑ n ≤ x d α ( n ) n 1 / 2 + i t ∣ 2 q d t ≪ ( log x ) 2 q ( α − 1 ) ( log log x ) 3 α q / 2 , \lim_{T\to\infty}\frac{1}{T}\int_T^{2T} \bigg|\sum_{n\leq x}\frac{d_\alpha(n)}{n^{1/2+it}}\bigg|^{2q} \mathrm{d}t \ll \frac{(\log x)^{2q(\alpha-1)}}{(\log\log x)^{3\alpha q/2}}, T → ∞ lim T 1 ∫ T 2 T n ≤ x ∑ n 1/2 + i t d α ( n ) 2 q d t ≪ ( log log x ) 3 α q /2 ( log x ) 2 q ( α − 1 ) , α ∈ ( 1 , 2 ) \alpha\in (1,2) α ∈ ( 1 , 2 ) q > 0 q>0 q > 0
引用
@article{arxiv.2604.05563,
title = {Partial sums of random multiplicative functions with supercritical divisor twists},
author = {Jad Hamdan},
journal= {arXiv preprint arXiv:2604.05563},
year = {2026}
}
备注
23 pages