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The function field abstract prime number theorem

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties (i) ∂(l) = 0,∂(p) > 0 for p∈P; (ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G; (iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)