偏 theta 函数零点模长的一个分离性质
经典分析与常微分方程
2019-05-10 v1 复变函数
摘要
我们考虑偏 theta 函数 θ ( q , z ) : = ∑ j = 0 ∞ q j ( j + 1 ) / 2 z j \theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j θ ( q , z ) := ∑ j = 0 ∞ q j ( j + 1 ) /2 z j z ∈ C z\in \mathbb{C} z ∈ C q ∈ C q\in \mathbb{C} q ∈ C 0 < ∣ q ∣ < 1 0<|q|<1 0 < ∣ q ∣ < 1 α 0 : = 3 / 2 π = 0.2756644477 … \alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots α 0 := 3 /2 π = 0.2756644477 … n ≥ 5 n\geq 5 n ≥ 5 ∣ q ∣ ≤ 1 − 1 / ( α 0 n ) |q|\leq 1-1/(\alpha _0n) ∣ q ∣ ≤ 1 − 1/ ( α 0 n ) k ≥ n k\geq n k ≥ n θ ( q , . ) \theta (q,.) θ ( q , . ) ξ k \xi _k ξ k ∣ q ∣ − k + 1 / 2 < ∣ ξ k ∣ < ∣ q ∣ − k − 1 / 2 |q|^{-k+1/2}<|\xi _k|<|q|^{-k-1/2} ∣ q ∣ − k + 1/2 < ∣ ξ k ∣ < ∣ q ∣ − k − 1/2 n − 1 n-1 n − 1 ≤ ∣ q ∣ − n + 1 / 2 \leq |q|^{-n+1/2} ≤ ∣ q ∣ − n + 1/2 q q q θ ( q , . ) \theta (q,.) θ ( q , . ) 0.4353184958 … ± i 0.1230440086 … 0.4353184958\ldots \pm i\, 0.1230440086\ldots 0.4353184958 … ± i 0.1230440086 … θ \theta θ − 5.963 … ± i 6.104 … -5.963\ldots \pm i\, 6.104\ldots − 5.963 … ± i 6.104 …
引用
@article{arxiv.1704.01901,
title = {A separation in modulus property of the zeros of a partial theta function},
author = {Vladimir Petrov Kostov},
journal= {arXiv preprint arXiv:1704.01901},
year = {2019}
}