标题： 素数、狄利克雷密度和本福德定律 显示英文标题

标题： Prime Numbers, Dirichlet Density, and Benford's Law

摘要： 素数以其与本福德定律的矛盾立场而闻名。 一方面，它们坚决拒绝遵循通常意义上的本福德定律，即以 d 为首数字的素数比例的正常密度；但另一方面，所有以 d 为首数字的素数子集的狄利克雷密度确实为 LOG(1 + 1/d)。 本文通过本福德定律领域内其他已知且确立的结果来展示并解释了狄利克雷密度结果的表面性，从概念上得出结论：尽管存在狄利克雷密度结果，素数不能被视为符合本福德定律。 此外，还概述了对素数数字行为的详细检查，显示出一种明显的数字发展模式，从小素数开始对低位数字略有偏好，到大素数时在极限情况下完全实现数字平等。 最后，推导出了素数对数密度的一个精确解析表达式，并证明它在宏观层面上始终呈上升趋势，这一观察也得到了实证验证。 显示英文摘要

摘要： The Prime Numbers are well-known for their paradoxical stand regarding Benford's Law. On one hand they adamantly refuse to obey the law of Benford in the usual sense, namely that of a normal density of the proportion of primes with d as the leading digit, yet on the other hand, the Dirichlet density for the subset of all primes with d as the leading digit is indeed LOG(1 + 1/d). In this article the superficiality of the Dirichlet density result is demonstrated and explained in terms of other well-known and established results in the discipline of Benford's Law, conceptually concluding that prime numbers cannot be considered Benford at all, in spite of the Dirichlet density result. In addition, a detailed examination of the digital behavior of prime numbers is outlined, showing a distinct digital development pattern, from a slight preference for low digits at the start for small primes, to a complete digital equality for large primes in the limit as the prime number sequence goes to infinity. Finally an exact analytical expression for the density of the logarithms of primes is derived and shown to be always on the rise at the macro level, an observation that is also confirmed empirically.