The abstract prime number theorem for function fields

Remarks on the infinite product representations of holomorphic functions

“…The proof of Proposition 1 is the same as that for the corresponding results about π(m) (see Proposition 1 in [4]) and is left to the reader.…”

“…Then, denote by g the logarithm of f for |z| < r, determined by exp g(z) = f (z) and g(0) = 0. We use the representation 3) and (4), respectively, to a formal power series with coefficients γ(n).…”

“…In [3], [4] we investigated the analytic behaviour of the product (6) in connection with applications for additive arithmetical semigroups.…”

Functorial orbit counting

“…[3] The results ( 18)-( 20) are shown in [12] to be consequences of Knopfmacher's Axiom A # in ( 17); ( 20) is due to Indlekofer [9]. [4] This is shown by Wehmeier [22].…”

Additive and multiplicative functions on arithmetical semigroups

“…Another nonequivalent axiom has been suggested by the first of the authors [5]. In [9] J. Knopfmacher gives an introductory analytic theory of semigroups satisfying Axiom A * , deeper problems he leaves as open questions (Chapter 12).…”