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Title:  Asymptotic Distribution of Beurling's Generalized Prime Numbers and Integers 
Author(s):  Zhang, WenBin 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This paper is a study by "elementary" and analytic methods of the asymptotic distribution of Beurling's generalized (henceforth g) prime numbers and integers Acta Math. 1937 . We call P = p(,i) (,i=1)('(INFIN)), where 1 ) (INFIN), a set of gprimes. The set of all products of gprimes is called the associated set of gintegers. Define summatory functions N(x), (psi)(x), (PI)(x) and M(x). 1. We show that with A > 0 and 0 < (theta) < 1. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) implies the Chebyshevtype estimates (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) This is a partial answer to a conjecture of Diamond Semin. Theo. Nom. Paris, 19741975 . 2. We generalize the famous Halasz theorem Acta Math. Acad. Sci. Hung. 1968 to gnumber systems. From this, we deduce that if N(x) = Ax + O(x log('(gamma))x), x > 1 holds with A > 0 and (gamma) > 1 then M(x) = o(x). This result, combined with Beurling's theorem and Diamond's example Ill. J. Math. 1970 , shows that the prime number theorem is not completely equivalent to the estimate M(x) = o(x). 3. It had been conjectured that de la Vallee Poussin's formula (PI)(x) = li x + O(x exp cSQRT.(log x) ) is essentially best possible for gprimes. We show that no example of Hall's type Ph.D. thesis Univ. of Illinois, 1967 can establish this conjecture. 4. We prove an Otype HardyLittlewoodKaramata tauberian theorem. With this, we give weak conditions on (PI)(x) that imply N(x) 0. 5. Let 
Issue Date:  1986 
Type:  Text 
Description:  144 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1986. 
URI:  http://hdl.handle.net/2142/71241 
Other Identifier(s):  (UMI)AAI8611005 
Date Available in IDEALS:  20141216 
Date Deposited:  1986 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

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