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http://dx.doi.org/10.4134/BKMS.2013.50.2.601

A CLASS OF ARITHMETIC FUNCTIONS ON PSL2(Z)  

Spiegelhalter, Paul (Department of Mathematics University of Illinois)
Zaharescu, Alexandru (Department of Mathematics University of Illinois)
Publication Information
Bulletin of the Korean Mathematical Society / v.50, no.2, 2013 , pp. 601-610 More about this Journal
Abstract
In [3] and [2], Atanassov introduced the two arithmetic functions $$I(n)=\prod_{p^{\alpha}{\parallel}n}\;p^{1/{\alpha}}\;and\;R(n)=\prod_{p^{\alpha}{\parallel}n}\;p^{{\alpha}-1}$$ called the irrational factor and the restrictive factor, respectively. Alkan, Ledoan, Panaitopol, and the authors explore properties of these arithmetic functions in [1], [7], [8] and [9]. In the present paper, we generalize these functions to a larger class of elements of $PSL_2(\mathbb{Z})$, and explore some of the properties of these maps.
Keywords
$PSL_2(\mathbb{Z})$; Farey fractions; Dirichlet series;
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1 E. Alkan, A. H. Ledoan, and A. Zaharescu, Asymptotic behavior of the irrational factor, Acta Math. Hungar. 121 (2008), no. 3, 293-305.   DOI   ScienceOn
2 K. T. Atanassov, Restrictive factor: definition, properties and problems, Notes Number Theory Discrete Math. 8 (2002), no. 4, 117-119.
3 K. T. Atanassov, Irrational factor: definition, properties and problems, Notes Number Theory Discrete Math. 2 (1996), no. 3, 42-44.
4 F. Boca and R. Gologan, On the distribution of the free path length of the linear flow in a honeycomb, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 1043-1075.   DOI   ScienceOn
5 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth, Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles, Oxford University Press, Oxford, 2008.
6 N. M. Korobov, Estimates of trigonometric sums and their applications, Uspehi Mat. Nauk 13 (1958), no. 4, 185-192.
7 A. Ledoan and A. Zaharescu, Real moments of the restrictive factor, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 4, 559-566.   DOI
8 L. Panaitopol, Properties of the Atanassov functions, Adv. Stud. Contemp. Math. (Kyungshang) 8 (2004), no. 1, 55-58.
9 P. Spiegelhalter and A. Zaharescu, Strong and weak Atanassov pairs, Proc. Jangjeon Math. Soc. 14 (2011), no. 3, 355-361.
10 E. C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1939.
11 E. C. Titchmarsh, The theory of the Riemann zeta-function, Edited and with a preface by D. R. Heath-Brown, The Clarendon Press Oxford University Press, New York, 1986.
12 I. M. Vinogradov, A new estimate of the function ${\zeta}$(1+it), Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 161-164.
13 A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutsche Verlag der Wissenschaften, Berlin, 1963.