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oai:www.ideals.illinois.edu:2142/86883 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Kevin Ford Hu, Yong 2015-09-28T15:19:59Z 2015-09-28T15:19:59Z 10000-01-01 2007 2007 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. Made available in DSpace on 2015-09-28T15:19:59Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3290250.pdf: 1758769 bytes, checksum: 59662d8e5eccfa807cce9e93bb2c97e1 (MD5) Previous issue date: 2007 Embargo set by: Seth Robbins for item 88164 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 71 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007. http://hdl.handle.net/2142/86883 (MiAaPQ)AAI3290250 eng Mathematics Localization of Divisors of Integers and of Some Arithmetic Functions text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.