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

Abstract

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, 273-290