素数分布分形涨落的普适特征
Universal Characteristics of Fractal Fluctuations in Prime Number Distribution
摘要：
素数的出现频率在单位数的间隔时间表现出自相似分形波动伴随着逆幂律形式的功率谱通用的动力系统在自然界中，如流体流动，股票市场波动，人口动态等的物理学的长程相关性所表现出的分形尚未确定。最近发展起来的一般系统理论可视化的涡连续体的分形导致从封闭的小尺度涡的综合平均值的大涡的增长，从而产生一个层次的涡环流，或相关的长程相关性的互连网络。模型预测如下：（i）分形的概率分布和功率谱遵循相同的逆幂律，这是一个黄金平均值的函数。预测的逆幂律分布是非常接近的统计正态分布的波动在两个标准偏差的平均分布。(ii)分形表示类量子混沌，因为方差谱表示概率密度分布，这是量子系统（如电子或光子）的特征。(ii)素数频率分布的分形波动表示潜在的连续统数场的准周期彭罗斯平铺模式的有序模式的自发组织。该模型的预测是在协议的概率分布和功率谱为不同的集合出现的素数在单位数间隔连续1000个数字的频率。前1000万个数字中的质数被用于研究。
英文摘要：
The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations, population dynamics, etc. The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualises the eddy continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a hierarchy of eddy circulations, or an inter-connected network with associated long-range correlations. The model predictions are as follows: (i) The probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the distribution. (ii) Fractals signify quantumlike chaos since variance spectrum represents probability density distribution, a characteristic of quantum systems such as electron or photon. (ii) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organisation of underlying continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime numbers in the first 10 million numbers were used for the study.
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