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Title:  Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations 
Author(s):  Spiegelhalter, Paul 
Director of Research:  Zaharescu, Alexandru 
Doctoral Committee Chair(s):  Berndt, Bruce C. 
Doctoral Committee Member(s):  Zaharescu, Alexandru; Hildebrand, A.J.; Boca, Florin 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Number theory
Dirichlet series Farey fractions 
Abstract:  K.T. Atanassov introduced the two arithmetic functions \[ I(n) = \prod_{\nu=1}^k p_\nu^{1/\alpha_\nu} \qquad \text{and}\qquad R(n) = \prod_{\nu=1}^k p_\nu^{\alpha_v  1} \] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations $I(n)^\alpha R(n)^\beta$ and characterize pairs $(\alpha,\beta)$ in order to measure how close $n$ is to being $k$power full or $k$power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complexvalued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems. 
Issue Date:  20140916 
URI:  http://hdl.handle.net/2142/50445 
Rights Information:  Copyright 2014 Paul Spiegelhalter 
Date Available in IDEALS:  20140916 20160922 
Date Deposited:  201408 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois