615 research outputs found
Benford's Law, Values of L-functions and the 3x+1 Problem
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.Comment: 25 pages, 1 figure; replacement of earlier draft (corrected some
typos, added more exposition, added results for characteristic polynomials of
unitary matrices
Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields
Given a fixed quadratic extension K of Q, we consider the distribution of
elements in K of norm 1 (denoted N). When K is an imaginary quadratic
extension, N is naturally embedded in the unit circle in C and we show that it
is equidistributed with respect to inclusion as ordered by the absolute Weil
height. By Hilbert's Theorem 90, an element in N can be written as
\alpha/\bar{\alpha} for some \alpha \in O_K, which yields another ordering of
\mathcal N given by the minimal norm of the associated algebraic integers. When
K is imaginary we also show that N is equidistributed in the unit circle under
this norm ordering. When K is a real quadratic extension, we show that N is
equidistributed with respect to norm, under the map \beta \mapsto \log| \beta |
\bmod{\log | \epsilon^2 |} where \epsilon is a fundamental unit of O_K.Comment: 19 pages, 2 figures, comments welcome
Sieving rational points on varieties
A sieve for rational points on suitable varieties is developed, together with
applications to counting rational points in thin sets, the number of varieties
in a family which are everywhere locally soluble, and to the notion of friable
rational points with respect to divisors. In the special case of quadrics,
sharper estimates are obtained by developing a version of the Selberg sieve for
rational points.Comment: 30 pages; minor edits (final version
Square-full polynomials in short intervals and in arithmetic progressions
We study the variance of sums of the indicator function of square-full
polynomials in both arithmetic progressions and short intervals. Our work is in
the context of the ring of polynomials over a finite field
of elements, in the limit . We use a recent
equidistribution result due to N. Katz to express these variances in terms of
triple matrix integrals over the unitary group, and evaluate them
- β¦