Benford’s law, values of \(L\)-functions and the \(3x+1\) problem. (English) Zbl 1139.11033

Summary: 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.


11K06 General theory of distribution modulo \(1\)
11B37 Recurrences
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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Online Encyclopedia of Integer Sequences:

Leading digits in 3x+1 chain starting with the number 1117065.