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arXiv 2307.06002math.NTmath-phmath.CVmath.MP
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Riemann zeta 函数邻域中晶格能量的非临界零点

On off-critical zeros of lattice energies in the neighborhood of the Riemann zeta function

Laurent Bétermin, Ladislav Šamaj, Igor Travěnec

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19 pages, 4 figures
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AI总结：

本文将 Riemann zeta 函数解释为一维晶格能量并引入交替间距参数 Δ，数值与解析揭示 Δ→1^- 时除临界零点外还存在实部趋于 -∞ 的无穷非临界零点。

摘要：

Riemann zeta 函数 ζ(s):=∑_{n=1}^∞ 1/n^s 可解释为晶格 Z 中每个点的能量，其中点之间通过 Riesz 势 1/r^s 进行两两相互作用。给定参数 Δ∈(0,1]，本文通过考虑周期性一维晶格的每点能量 E(s,Δ) 来推广这一物理模型；该晶格将最近邻粒子之间的距离交替设为 2/(1+Δ) 和 2Δ/(1+Δ)，并使晶格密度独立于 Δ 而保持为一。该能量在 Δ=1 时显然满足 E(s,1)=ζ(s)，可以容易地表示为 Riemann zeta 函数与 Hurwitz zeta 函数的组合，并解析延拓到穿孔 s 平面 C\{1}。本文对由 E(ρ,Δ)=0 定义的能量零点 {ρ=ρ_x+iρ_y} 进行数值研究。数值结果表明，在 Riemann 极限 Δ→1^- 下，这些零点既包括预期的、满足 Re(ρ_x)=1/2 的 Riemann zeta 函数临界零点，也包括与 Riemann 假设相比出乎意料的无穷系列非临界零点。对这些非临界零点的解析处理表明，其虚部等距，而当 Δ→1^- 时其实部对数发散至 -∞，也就是说，它们在 Riemann 的 Δ=1 处变得不可见。

英文摘要：

The Riemann zeta function $ζ(s):= \sum_{n=1}^{\infty} 1/n^s$ can be interpreted as the energy per point of the lattice $\mathbb{Z}$, interacting pairwisely via the Riesz potential $1/r^s$. Given a parameter $Δ\in (0,1]$, this physical model is generalized by considering the energy per point $E(s,Δ)$ of a periodic one-dimensional lattice alternating the distances between the nearest-neighbour particles as $2/(1+Δ)$ and $2Δ/(1+Δ)$, keeping the lattice density equal to one independently of $Δ$. This energy trivially satisfies $E(s,1)=ζ(s)$ at $Δ=1$, it can be easily expressed as a combination of the Riemann and Hurwitz zeta functions, and extended analytically to the punctured $s$-plane $\mathbb{C} \setminus \{ 1\}$. In this paper, we perform numerical investigations of the zeros of the energy $\{ ρ=ρ_x+{\rm i}ρ_y\}$, which are defined by $E(ρ,Δ)=0$. The numerical results reveal that in the Riemann limit $Δ\to 1^-$ theses zeros include the anticipated critical zeros of the Riemann zeta function with $\Re(ρ_x)=\frac{1}{2}$ as well as an unexpected -- comparing to the Riemann Hypothesis -- infinite series of off-critical zeros. The analytic treatment of these off-critical zeros shows that their imaginary components are equidistant and their real components diverge logarithmically to $-\infty$ as $Δ\to 1^-$, i.e., they become invisible at the Riemann's $Δ=1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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