On Beurling's Theory of Generalized Numbers
Following A. Beurling (Acta Math. 68 (1937) 255-291), we consider a set of generalized prime numbers $P=\{1<p\sb1\le p\sb2\le\...\}$ and the set of generalized integers $N=\{n\sb1=1\le n\sb2\le\...\}$ generated by P. We let $N(x)$ be the counting function of the set N. In this thesis we give continuous versions of generalized number systems considered by R. S. Hall (Proc. Amer. Math. Soc. 40 (1973) 79-82). We provide an explicit calculation of the associated zeta function, which in turn allows us to obtain an expression for $N(x)$ with several terms rather than just an O-term for the error. We construct examples that illustrate a theorem of W.-B. Zhang (Illinois J. Math. 31 (1987) 645-664) about the mean value of the generalized Mobius function. We construct a continuous number system for which the error term in the integer counting function $N(x)$ oscillates as much as the de la Vallee Poussin error for the Prime Number Theorem. A simplification in a theorem of H. G. Diamond is offered. Finally we estimate the integral of a measure obtained by interpolating the integer counting measure and (essentially) the Lebesgue measure.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign