University of Illinois Urbana-Champaign

Asymptotic Distribution of Beurling's Generalized Prime Numbers and Integers

Zhang, Wen-Bin

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Description

Title
Asymptotic Distribution of Beurling's Generalized Prime Numbers and Integers
Author(s)
Zhang, Wen-Bin
Issue Date
1986
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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 g-primes. The set of all products of g-primes is called the associated set of g-integers. 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 Chebyshev-type estimates
  • This is a partial answer to a conjecture of Diamond Semin. Theo. Nom. Paris, 1974-1975 .
  • 2. We generalize the famous Halasz theorem Acta Math. Acad. Sci. Hung. 1968 to g-number 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 g-primes. We show that no example of Hall's type Ph.D. thesis Univ. of Illinois, 1967 can establish this conjecture.
  • 4. We prove an O-type Hardy-Littlewood-Karamata tauberian theorem. With this, we give weak conditions on (PI)(x) that imply N(x) 0.
  • 5. Let
Graduation Semester
1986
Type of Resource
text
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http://hdl.handle.net/2142/71241

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