Localization of Divisors of Integers and of Some Arithmetic Functions

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 u-dense 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 2-dense.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Hu, Yong

Contributors dc:contributor

• Kevin Ford

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3290250

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86883