Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2013; 50(2): 601-610

Printed March 31, 2013

https://doi.org/10.4134/BKMS.2013.50.2.601

Copyright © The Korean Mathematical Society.

A class of arithmetic functions on $\text{PSL}_2(\mathbb{Z})$

Paul Spiegelhalter and Alexandru Zaharescu

University of Illinois, University of Illinois

Abstract

In \cite{MR1429354} and \cite{MR2021015}, Atanassov introduced the two arithmetic functions \[ I(n) = \prod_{p^\alpha || n} p^{1/\alpha} ~ \text{ and }~ R(n) = \prod_{p^\alpha || n} p^{\alpha -1}\] called the irrational factor and the restrictive factor, respectively. Alkan, Ledoan, Panaitopol, and the authors explore properties of these arithmetic functions in \cite{MR2452808}, \cite{MR2647199}, \cite{MR2022035} and \cite{pre05949797}. In the present paper, we generalize these functions to a larger class of elements of $\text{PSL}_2(\mathbb{Z})$, and explore some of the properties of these maps.

Keywords: $\text{PSL}_2(\mathbb{Z})$, Farey fractions, Dirichlet series

MSC numbers: Primary 11N37; Secondary 11B57