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Zhang, Wen-Bin 2014-12-16T06:18:15Z 2014-12-16T06:18:15Z 10000-01-01 1986 1986 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 Made available in DSpace on 2014-12-16T06:18:15Z (GMT). No. of bitstreams: 1 8611005.pdf: 2474875 bytes, checksum: f17985ca0602be686a228562c8aab317 (MD5) Previous issue date: 1986 Embargo set by: Seth Robbins for item 71407 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 144 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1986. http://hdl.handle.net/2142/71241 (UMI)AAI8611005 Mathematics Asymptotic Distribution of Beurling's Generalized Prime Numbers and Integers text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.