Number Theoretic Density and Logical Limit Laws

Number Theoretic Density
and
Logical Limit Laws

Stanley N. Burris
E-mail: snburris@math.uwaterloo.ca

Mathematical Surveys and Monographs Series
© American Mathematical Society
ISBN 0-8218-2666-2
Available from the AMS bookstore (www.ams.org)

Table of Contents	[ PDF ] [ PS ]
Preface	[ PDF ] [ PS ]
Overview	[ PDF ] [ PS ]

Errata to Text

UPDATES

Richard Warlimont has answered the First Conjecture in the affirmative in his preprint On the zeta function of an arithmetical semigroup.
Jason Bell has answered the Second and Third Conjectures in the affirmative.
Karen Yeats has made significant (and in key respects definitive) progress on Problems 5.20 and 11.25 in her paper Asymptotic density in combined number systems, New York J. Math 8(2002), 63-83.
In A true multiplicative analogue of Schur's Tauberian Theorem, Canad. Math. Bull. 46 no.3 (2003), 473-480, she improves a result in my book.
Reference [33] John Knopfmacher and Richard Warlimont, Arithmetical semigroups related to trees and polyhedra, II -- Maps on surfaces will appear in Math. Nachr.
Reference [6], Jason P. Bell, Sufficient conditions for zero-one laws, Trans. Amer. Math. Soc. 354 (2002), 613--630.
Reference [55] is now a paper by Richard Warlimont, About the abscissa of convergence of the Zeta function of a multiplicative arithmetical semigroup. (to appear in Quaestiones Mathematicae).
For further information on the discovery of the asymptotics of the coefficients of exp(a/(1-x)^rho) see the following two papers of E.M. Wright:
- The coefficients of certain power series. J. London Math. Soc. 7 256-262 (1932)
- On the coefficients of power series having exponential singularities. (Second paper). J. London Math. Soc. 24 304-309 (1950).
I am indebted to Andreas Weiermann for relaying this information (provided by Phillipe Flajolet) to me.

Some Outstanding Open Problems

Find an example of a number system in RT_rho whose generating function diverges at rho with the property that some partition set does not have asymptotic density.
Find an example of a number system in RV_alpha whose generating function diverges at alpha with the property that some partition set does not have asymptotic density.
Give a multiplicative analog of the results of Woods.

My Related Preprint(s) (1) (with Jason P. Bell) Partition Identities I. Sandwich Theorems and Logical 0--1 Laws [ PS ] [ DVI ] [ PDF ] (2) (with Jason P. Bell) Partition Identities II. The Results of Bateman and Erdos [ PS ] [ DVI ] [ PDF ] (3) (with Jason P. Bell) Asymptotics for Logical Limit Laws. When the Growth of the Components is in an RT Class. Trans. Amer. Math. Soc. 355 (2003), 3777--3794. [ PS ] [ DVI ] [ PDF ] (4) (with Jason P. Bell ) Dirichlet Density Extends Global Asymptotic Density in Multiplicative Systems [ PS ] [ DVI ] [ PDF ] (5) (with Karen Yeats) Admissible Dirichlet Series [ PS ] [ DVI ] [ PDF ]
Jason's Homepage J. Bell's Publication List Karen's Homepage K. Yeat's Publication List

My Related Preprint(s)

(1) (with Jason P. Bell)
Partition Identities I.
Sandwich Theorems and Logical 0--1 Laws
[ PS ] [ DVI ] [ PDF ]
(2) (with Jason P. Bell)
Partition Identities II.
The Results of Bateman and Erdos
[ PS ] [ DVI ] [ PDF ]
(3) (with Jason P. Bell)
Asymptotics for Logical Limit Laws.
When the Growth of the Components is in an RT Class.
Trans. Amer. Math. Soc. 355 (2003), 3777--3794.
[ PS ] [ DVI ] [ PDF ]
(4) (with Jason P. Bell )
Dirichlet Density Extends Global Asymptotic Density in Multiplicative Systems
[ PS ] [ DVI ] [ PDF ]
(5) (with Karen Yeats)
Admissible Dirichlet Series
[ PS ] [ DVI ] [ PDF ]

Jason's Homepage J. Bell's Publication List

Karen's Homepage K. Yeat's Publication List