African Journals Online
Quaestiones Mathematicae

VOLUME 24 ISSUE 3 (2001)

Quaestiones Mathematicae 24(3) 2001, 261-262
© 2001 NISC Pty Ltd, www.nisc.co.za  

John Knopfmacher - A Mathematical Biography

Doron Lubinsky

The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected] 

Keywords: John Knopfmacher, biography

Quaestiones Mathematicae 24(3) 2001, 263-267
© 2001 NISC Pty Ltd, www.nisc.co.za  

John Knopfmacher - Mathematical and Other Memories

Arnold Knopfmacher

The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected] 

Keywords: John Knopfmacher

Quaestiones Mathematicae 24(3) 2001, 269-272
© 2001 NISC Pty Ltd, www.nisc.co.za  

John Knopfmacher and Additive Arithmetical Semigroups

Richard Warlimont

The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected] 

Keywords: John Knopfmacher, additive arithmetical semigroups

Quaestiones Mathematicae 24(3) 2001, 273-290
© 2001 NISC Pty Ltd, www.nisc.co.za  

John Knopfmacher, [] Analytic Number Theory, and the Theory of Arithmetical Functions

Wolfgang Schwarz

Department of Mathematics, Johann Wolfgang Goethe-Universität, Robert-Mayer-Straße 10, D 60054 Frankfurt am Main, Germany.

e-mail: [email protected] 

In this paper some important contributions of John Knopfmacher to ' Analytic Number Theory' are described. This theory investigates semigroups with countably many generators (generalized 'primes'), with a norm map (or a 'degree map'), and satisfying certain conditions on the number of elements with norm less than x (Axiom A resp. Axiom $\A^{#}$), and 'arithmetical' functions defined on these semigroups.
It is tried to show some of the impact of John Knopfmachers ideas to the future development of number theory, in particular for the topics 'arithmetical functions' and asymptotics in additive arithmetical semigroups.

Mathematics Subject Classification (1991): 11-02, 11N45, 11N80

Keywords: John Knopfmacher, analytic number theory, arithmetical functions, research exposition, asymptotic, counting functions, algebraic structures, topological structures, generalized primes and integers, Ramanujan expansions, analytic number theory, number theory, semigroup, generators, primes, additive arithmetical semigroups, additive arithmetical semigroup, arithmetical semigroups

Quaestiones Mathematicae 24(3) 2001, 291-307
© 2001 NISC Pty Ltd, www.nisc.co.za  

Recent Developments and Applications of Analytic Number Theory

John Knopfmacher

Centre for Applicable Analysis and Number Theory, University of Witwatersrand, Johannesburg, South Africa.

The aim of this survey is to outline some basic concepts and results of "" analytic number theory, with an emphasis on applications to concrete systems arising in a variety of mathematical fields. Some remarks on more recent investigations are also included.

Mathematics Subject Classification (1991): 11-02, 11N, 11T55

Keywords: analytic number theory, research exposition, multiplicative number theory, arithmetic theory of polynomial rings over finite fields, arithmetical semigroups, semisimple finite rings, Lie, symmetric Riemannian manifolds, finite topological spaces, finite graphs, zeta functions, number theory

Quaestiones Mathematicae 24(3) 2001, 309-322
© 2001 NISC Pty Ltd, www.nisc.co.za  

Contributions to Analytic Number Theory

Lutz G Lucht

Institut für Mathemätik , Technische Universitat Clausthal, 38678 Clausthal-Zellereld, Germany.

e-mail: [email protected] 

This paper reports on some recent contributions to the theory of multiplicative arithmetic semigroups, which have been initiated by John Knopfmacher's work on analytic number theory. They concern weighted inversion theorems of the Wiener type, mean-value theorems for multiplicative functions, and, Ramanujan expansions.

Mathematics Subject Classification (1991): 11A25, 11N37

Keywords: analytic number theory, arithmetic functions, related numbers, inversion formulas, Dirichlet algebras, arithmetic semigroups, weighted inversion theorems, semigroup, John Knopfmacher, analytic number theory, number theory, multiplicative function, Ramanujan expansions, expansions

Quaestiones Mathematicae 24(3) 2001, 323-333
© 2001 NISC Pty Ltd, www.nisc.co.za  

Analytic and Probabilistic Theory of Additive Arithmetical Semigroups

Wen-Bin Zhang

Department of Mathematics, Iniversity of Illinois, 1409 West Green Street, Urbana, IL 61801, USA.

e-mail: [email protected] 

This is a short survey of the forthcoming book Number Theory Arising From Finite Fields - analytic and probabilistic theory. We give details of a number of the main theorems in the book. These are prime number theorems, mean-value theorems of multiplicative functions, infinitely divisible distributions and central limit theorems. We also highlight new research directions.

Mathematics Subject Classification (1991): Primary: 11N60; Secondary: 11T55, 30B30, 60F05

Keywords: Additive arithmetical semigroup, number theorem, mean-value theorem, normal distribution

Quaestiones Mathematicae 24(3) 2001, 335-347
© 2001 NISC Pty Ltd, www.nisc.co.za  

Distribution of Multiplicative Functions Defined on Semigroups

K -H Indlekofer¹ and E Manstavicius²

¹ Universitat Paderborn, Fachbereich Mathematik und Informatik, Warburger Strasse 100, D-33098 Paderborn BRD.

e-mail: [email protected] 

² Vilnius University, Department of Mathematics and Informatics, Naugarduko str. 24, Vilnius, LT2600 Lithuania.

e-mail: [email protected] 

The value distribution problem for real-valued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of number-theoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.

Quaestiones Mathematicae 24(3) 2001, 349-354
© 2001 NISC Pty Ltd, www.nisc.co.za  

Number Systems and Logical Limit Laws

Stanley Burris

Deartment of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

e-mail: [email protected] 

This paper gives a brief overview of the developments that led to writing the book [5], Number Theoretic Density and Logical Limit Laws, simply called Density below, and describes the significant role played by the work and correspondence of John Knopfmacher.

Mathematics Subject Classification (1991): 11N45, 11P82, 03C13

Keywords: number systems, logical limit laws, additive number system, finite structures, asymptotic, counting functions, algebraic structures, topological structures, analytic theory of partitions, distribution, multiplicative function, additive arithmetical semigroup, semigroup, additive function, general, density, John Knopfmacher

Quaestiones Mathematicae 24(3) 2001, 355-362
© 2001 NISC Pty Ltd, www.nisc.co.za  

About the Radius of Convergence of the Zeta Function of an Additive Arithmetical Semigroup

Richard Warlimont

The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected] 

Let (G, $\partial$) be an additive arithmetical semigroup in John Knopfmacher's sense with f(n), g(n) denoting the number of all elements, prime elements of G with degree $\partial$ = n.
It is shown that if $\lim_{n \to \infty} sup g(n)/f(n) < 1$ then the zeta function of (G, $\partial$) which is a power series has a positive radius of convergence.

Mathematics Subject Classification (1991): 11N45

Keywords: radius of convergence, zeta function, additive arithmetical semigroup, asymptotic, counting functions, algebraic structures, topological structures, density, logical limit laws, John Knopfmacher, semigroup, convergence

Quaestiones Mathematicae 24(3) 2001, 363-372
© 2001 NISC Pty Ltd, www.nisc.co.za  

About the Abscissa of Convergence of the Zeta Function of a Multiplicative Arithmetical Semigroup

Richard Warlimont

The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected] 

Let (G, $\|\cdot\|$) be a multiplicative arithmetical semigroup in John Knopfmacher's sense with F(x), G(x) denoting the number of all elements, prime elements of G with norm $\mid \cdot \mid \leq x$.
It is shown that if $\lim_{x \to \infty} sup G(x)/F(x) < 1$ then the zeta function of (G, \| \cdot \|$) which is a Dirichlet series, has a finite abscissa of convergence.

Mathematics Subject Classification (1991): 11N45

Keywords: zeta function, topological structures, algebraic structures, asymptotic, counting functions, abscissa, multiplicative arithmetical semigroup, additive arithmetical semigroup, semigroup, John Knopfmacher, radius of convergence, convergence

Quaestiones Mathematicae 24(3) 2001, 373-391
© 2001 NISC Pty Ltd, www.nisc.co.za  

Arithmetically Related Ideal Topologies and the Infinitude of Primes

Stefan Porubský

Department of Mathematics, Institute of Chemical Technology, Technická 5, 166 28 Prague 6, Czech Republic.

e-mail:  [email protected] 

The late J. Knopfmacher and the author [12] have studied some ties between arithmetic properties of the multiplicative structure of commutative rings with identity and the topologies induced by some coset classes. In the present communication it is shown that the ideas used there are capable of a further extension. Namely, replacing the ideal structure of commutative rings by generalized ideal systems, the so called x-ideals, conditions implying the existence of infinitely many prime x-ideals are found using topologies induced by cosets of x-ideals. This leads to new variants of Fürstenberg topological proof of the infinitude of prime numbers not depending on the additive structure of the underlying integers or commutative rings with identity. as a byproduct we give new proofs of the infinitude of primes based on tools taken from commutative algebra.

Mathematics Subject Classification (1991): 11N80, 11N25, 11A41, 11T99, 13A15, 20M25

Keywords: x-ideal, topological semigroup, ideal topology, infinitude of primes, generalized primes and integers, distribution, integers, specified multiplicative constraints, primes, ideals, multiplicative ideal theory, semigroup rings, multiplicative semigroups of rings, multiplicative arithmetical semigroup, semigroup, John Knopfmacher, zeta function, abscissa, convergence, commutative, rings, ring, identities, algebra

Quaestiones Mathematicae 24(3) 2001, 393-401
© 2001 NISC Pty Ltd, www.nisc.co.za  

Knopfmacher Expansions in Number Theory

S Kalpazidou¹ and C Ganatsiou²

¹ Aristotle University, School of Sciences, Department of Mathematics, 54006 Thessaloniki, Greece.

² Technological Educational Institute of Larissa, School of Technological Applications, Larissa, Greece.

e-mail: [email protected] 

We survey in this paper J. Knopfmacher results on number theoretical expansions as they arised from our common collaboration.

Mathematics Subject Classification (1991): 11K55, 60J10, 60G10, 60K99

Keywords: metric theory, algorithms, expansions, measure dimension, Hausdorff dimension, Knopfmacher expansions, number theory, Markov chains, discrete parameter, commutative, rings, ring, identities, x-ideal, integers, infinitude of primes, primes, algebra

Quaestiones Mathematicae 24(3) 2001, 403-416
© 2001 NISC Pty Ltd, www.nisc.co.za  

q-Engel Series Expansions and Slaters Identities

George E Andrews¹, Arnold Knopfmacher², Peter Paule³ and Helmut Prodinger⁴

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.

e-mail: [email protected] 

² The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected]  ; http://www.wits.ac.za/sciencw/number_theory/arnold.htm 

³ Research Institute for Symbolic Computation, Johannes Kepler University Linz, A-4040 Linz, Austria.

e-mail: [email protected] 

⁴ The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg 2050, South Africa.

e-mail: [email protected]  ; http://www.wits.ac.za/helmut/index.htm 

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial role.

Mathematics Subject Classification (1991): Primary: 11; Secondary: 11.

Keywords: expansions, q-Engel series, Slaters identities, Engel series, q-series, Al-Salam polynomials, Ismail polynomials, identities, Santos polynomials, number theory, John Knopfmacher, partition, study, polynomials