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