Benford's Law, Values of L-functions and the 3x+1 Problem

Abstract

We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford.

25 pages, 1 figure; replacement of earlier draft (corrected some typos, added more exposition, added results for characteristic polynomials of unitary matrices)

References in corpus (2)

• The Modulo 1 Central Limit Theorem and Benford's Law for Products
• On small values of the Riemann zeta-function on the critical line and gaps between zeros

Cited by in corpus (18)

• The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)
• The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)
• Order Statistics and Benford's Law
• The Modulo 1 Central Limit Theorem and Benford's Law for Products
• Chains of distributions, hierarchical Bayesian models and Benford's Law
• The Weibull Distribution and Benford's Law
• Benford's Law and Continuous Dependent Random Variables
• Stochastic Models for the 3x+1 and 5x+1 Problems
• Benford's law from 1881 to 2006
• Benford's law for the $3x+1$ function
• A characterization of Benford's Law in discrete-time linear systems
• The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law
• On Logarithmically Benford Sequences
• Explicit bounds for the approximation error in Benford's law
• Some elements of a possible demonstration of the Collatz conjecture
• Most linear flows on ${\mathbb{R}}^d$ are Benford
• Benford's Law For Coefficients of Modular Forms and Partition Functions
• An E-sequence approach to the 3x + 1 problem