### 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