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An elemental Erdős-Kac theorem for algebraic number fields


Author: Paul Pollack
Journal: Proc. Amer. Math. Soc. 145 (2017), 971-987
MSC (2010): Primary 11N37; Secondary 11R27, 11R29
DOI: https://doi.org/10.1090/proc/13476
Published electronically: November 29, 2016
MathSciNet review: 3589297
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Abstract: Fix a number field $ K$. For each nonzero $ \alpha \in \mathbf {Z}_K$, let $ \nu (\alpha )$ denote the number of distinct, nonassociate irreducible divisors of $ \alpha $. We show that $ \nu (\alpha )$ is normally distributed with mean proportional to $ (\log \log \vert N(\alpha )\vert)^{D}$ and standard deviation proportional to $ (\log \log {\vert N(\alpha )\vert})^{D-1/2}$. Here $ D$, as well as the constants of proportionality, depend only on the class group of $ K$. For example, for each fixed real $ \lambda $, the proportion of $ \alpha \in \mathbf {Z}[\sqrt {-5}]$ with

$\displaystyle \nu (\alpha ) \le \frac {1}{8}(\log \log {N(\alpha )})^2 + \frac {\lambda }{2\sqrt {2}} (\log \log {N(\alpha )})^{3/2} $

is given by $ \frac {1}{\sqrt {2\pi }} \int _{-\infty }^{\lambda } e^{-t^2/2}\, \mathrm {d}t$. As further evidence that ``irreducibles play a game of chance'', we show that the values $ \nu (\alpha )$ are equidistributed modulo $ m$ for every fixed $ m$.

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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: pollack@uga.edu

DOI: https://doi.org/10.1090/proc/13476
Keywords: Erd\H{o}s--Kac theorem, Davenport constant, number field, irreducible element
Received by editor(s): March 22, 2016
Published electronically: November 29, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society