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2014-07

A scaling property of Farey fractions

by Kunik, Matthias.


Series: 2014-07, Preprints

MSC:
11B57 Farey sequences; the sequences ${1^k, 2^k, cdots}$
11M06 $zeta (s)$ and $L(s, chi)$
42B10 Fourier and ~Fourier-Stieltjes transforms and other transforms of Fourier type
42B30 $H^p$-spaces

Abstract:
The Farey sequence of order $n$ consists of all reduced fractions $a/b$ between $0$ and $1$
with positive denominator $b$ less or equal to $n$. The sums of the inverse denominators $1/b$
of the Farey fractions in certain scaling intervals with one fixed rational bound have a simple main term,
but their fluctuations are determined by an interesting sequence of polygonal functions $f_n$.
For $n to infty$ we also obtain a certain limit function, which is independent of $a/b$
and which describes the behaviour of the functions $f_n$ in the vicinity of any fixed
fraction. This result can be obtained by using only elementary methods.
We also study the asymp-totic behaviour of this limit function
by using analytical properties of the Riemann zeta function.

Keywords:
Farey sequences, Riemann zeta function, Fourier analysis, Hardy spaces