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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.

##### MSC:
 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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