JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY,AND THE THEORY OF ARITHMETICAL FUNCTIONS

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

In this paper some important contributions of John Knopfmacher to ‘Abstract 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’.