×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI arXiv