2026-09-21T13:36:44Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/50445 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Zaharescu, Alexandru Berndt, Bruce C. Zaharescu, Alexandru Hildebrand, A.J. Boca, Florin Spiegelhalter, Paul 2014-09-16T17:17:33Z 2014-09-16T17:17:33Z 2016-09-22T20:59:27Z 2014-08 2014-09-16 2014-08 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 complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-10T15:12:50Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Spiegelhalter_Paul.pdf: 1417493 bytes, checksum: 297993f16596438f7ff61e60ab327690 (MD5) Made available in DSpace on 2014-09-16T17:17:33Z (GMT). No. of bitstreams: 2 Paul_Spiegelhalter.pdf: 1417483 bytes, checksum: dcbc91a7dc214b6a293b5478cf5e61ff (MD5) license.txt: 4068 bytes, checksum: 309586c51a412426a012945708741f0c (MD5) Embargo set by: Seth Robbins for item 50556 Lift date: 2016-09-16T17:18:17Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system U of I Only Restriction Lifted for Item 50556 on 2016-09-22T20:59:27Z. http://hdl.handle.net/2142/50445 en Copyright 2014 Paul Spiegelhalter Number theory Dirichlet series Farey fractions Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD