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Title:  Sums of Multiplicative Functions Over Integers Without Large Prime Factors and Related Differential Difference Equations 
Author(s):  Song, Joung Min 
Doctoral Committee Chair(s):  Diamond, Harold G. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Integers without large prime factors arise naturally in various areas of number theory. Counting functions of such integers have been the subject of numerous past studies. Traditionally, Psi(x, y) denotes the number of positive integers up to x, all of whose prime divisors are less than or equal to y. The estimate of Psi(x, y) is given in terms of Dickman's famous function rho(u), where u = log x/log y. In this thesis, we study generalizations of Psi(x, y): Let S(x, y) denote the set of positive integers up to x, all of whose prime divisors are at most y, and we consider sums M( x, y) = n∈Sx,y h(n), and m( x, y) = n∈Sx,y h(n)/n, where h(n) is a nonnegative multiplicative function satisfying a set of weak summatory conditions on primes and prime powers. When h(n) ≡ 1 we have M( x, y) = Psi(x, y). The estimates of M(x, y) and m(x, y) are expressed in terms of a family of continuous functions that are related to Dickman's function. We obtain these results by an inductive argument that was inspired by A. Hildebrand's estimate of Psi(x, y), starting with initial estimates of m(x, y) and M(x, y) of H. Halberstam. 
Issue Date:  2000 
Type:  Text 
Language:  English 
Description:  82 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 2000. 
URI:  http://hdl.handle.net/2142/86999 
Other Identifier(s):  (MiAaPQ)AAI9971195 
Date Available in IDEALS:  20150928 
Date Deposited:  2000 
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Dissertations and Theses  Mathematics

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