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Title:  Localization of Divisors of Integers and of Some Arithmetic Functions 
Author(s):  Hu, Yong 
Doctoral Committee Chair(s):  Kevin Ford 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  We investigate several problems related to the multiplicative structure of integers. First, we determine the order of magnitude of the function H2(x, y, z), the number of positive integers n ≤ x having exactly two divisors in ( y, z], in the range of y10 ≤ z ≤ x1/3, and that of the function H3(x, y, z), the number of positive integers n ≤ x having exactly three divisors in ( y, z], in the range of y10 ≤ z ≤ x1/4. Next, we introduce a new function H1,1(x, y, z 1, z2), the number of positive integers n ≤ x having one divisor in (y, z 1] and one divisor in (z1, z 2] and study its order of magnitude. Finally, we investigate problems about localization of divisors of the Euler function &phis;(n) and the Carmichael function lambda(n). We say that a positive integer m is udense if whenever 1 ≤ y ≤ m, there is a divisor of m in the interval (y, uy]. We show that for > x integers n ≤ x, both &phis;(n) and lambda(n) are 2dense. 
Issue Date:  2007 
Type:  Text 
Language:  English 
Description:  71 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2007. 
URI:  http://hdl.handle.net/2142/86883 
Other Identifier(s):  (MiAaPQ)AAI3290250 
Date Available in IDEALS:  20150928 
Date Deposited:  2007 
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Dissertations and Theses  Mathematics

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