标题： 分形拟合到黎曼零点 显示英文标题

标题： Fractal fits to Riemann zeros

摘要： 吴和斯普龙格（Phys. Rev. E 48, 2595 (1993)）使用一维局部势模型再现了前500个非平凡的黎曼零点。他们得出结论——同样地，范齐尔和哈钦森（Phys. Rev. E 67, 066211 (2003)）——势具有维度d=3/2的分形结构。我们通过贝里和刘易斯的A(x,g)交替符号正弦级数分形来建模势的不光滑波动部分。设定d=3/2，我们估计频率参数（gamma），以及我们引入的整体缩放参数（sigma）。我们寻找使最低n个本征值的最小二乘拟合S_{n}(gamma,sigma)最小的参数对（gamma,sigma）——这些本征值是通过用试用势（平滑部分加上不平滑部分）求解一维定态（非分形）薛定谔方程得到的——与最低n个黎曼零点的拟合，其中n=25。对于我们研究的其他情况，n=50和75，我们简单地设置sigma=1。获得的拟合结果与仅使用吴-斯普龙格势的平滑部分而没有任何分形补充的情况进行比较。在我们的（非优化、计算受限）搜索过程中发现了一些有限的改进——5.7261对比6.39207（n=25），11.2672对比11.7002（n=50）和16.3119对比16.6809（n=75）。在gamma=3及其平方9附近，改进相对较强。此外，我们将吴-斯普龙格半经典框架扩展到包括来自黎曼-冯·曼戈尔德公式的高阶修正（超出主要主导项）到平滑势中。 显示英文摘要

摘要： Wu and Sprung (Phys. Rev. E 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded -- and similarly van Zyl and Hutchinson (Phys. Rev. E 67, 066211 (2003)) -- that the potential possesses a fractal structure of dimension d=3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting d=3/2, we estimate the frequency parameter (gamma), plus an overall scaling parameter (sigma) we introduce. We search for that pair of parameters (gamma,sigma) which minimizes the least-squares fit S_{n}(gamma,sigma) of the lowest n eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the lowest n Riemann zeros for n =25. For the additional cases we study, n=50 and 75, we simply set sigma=1. The fits obtained are compared to those gotten by using just the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement -- 5.7261 vs. 6.39207 (n=25), 11.2672 vs. 11.7002 (n=50) and 16.3119 vs. 16.6809 (n=75) -- is found in our (non-optimized, computationally-bound) search procedures. The improvements are relatively strong in the vicinities of gamma=3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher-order corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential.