Riemann零点的推广分形拟合
Extended Fractal Fits to Riemann Zeros
摘要：
我们扩展到前300个黎曼零点，我们在arXiv：math-ph/0606005中报告的分析形式，其中最大的研究涉及前75个零点。再次，我们模型的非光滑波动的Wu-Sprung势的一部分，再现黎曼零点，由交替符号的正弦级数分形Berry和Lewis A（x，g）。将分形维数设置为3/2。我们估计频率参数（g）加上引入的总缩放参数（s）。我们搜索这对参数（g，s），最小化最低的300个特征值的最小二乘拟合-通过求解一维固定（非分形）薛定谔方程与审判潜力（光滑加上非光滑部分）-到前300个黎曼零点。我们在矩形0 < s < 3，0 < g < 25内随机采样值。得到的拟合进行比较，通过简单地使用无任何分形补充的吴-Sprung势的光滑部分。一些有限的改善再次被发现。有两个（主要和次要）相当不同的子域，其中值给予改善拟合集中。
英文摘要：
We extend to the first 300 Riemann zeros, the form of analysis reported by us in arXiv:math-ph/0606005, in which the largest study had involved the first 75 zeros. Again, we model the nonsmooth fluctuating part of the Wu-Sprung potential, which reproduces the Riemann zeros, by the alternating-sign sine series fractal of Berry and Lewis A(x,g). Setting the fractal dimension equal to 3/2. we estimate the frequency parameter (g), plus an overall scaling parameter (s) introduced. We search for that pair of parameters (g,s) which minimizes the least-squares fit of the lowest 300 eigenvalues -- obtained by solving the one-dimensional stationary (non-fractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) -- to the first 300 Riemann zeros. We randomly sample values within the rectangle 0 < s < 3, 0 < g < 25. The fits obtained are compared to those gotten by using simply the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement is again found. There are two (primary and secondary) quite distinct subdomains, in which the values giving improvements in fit are concentrated.
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