As is well known, the abstract prime number theorem for an algebraic function field is proved in the context of an additive arithmetic semigroup since the concept of the latter was introduced by Knopfmacher, [8]. Thus it is essentially a theorem about prime elements in additive arithmetic semigroups. We recall that an additive arithmetic semigroup G is, by definition, a free commutative semigroup with identity element 1 such that G has a countable free generating set P of "primes" p and such that G admits an integer-valued degree mapping partial deriv: G → N ∪ {0} satisfying (1) partial deriv(1) = 0 and partial deriv(p) > 0 for all p ε P, (2) partial deriv(ab) = partial deriv(a) + partial deriv(b) for all a,b ε G, and (3) the total number G(n) of elements of degree n in G is finite for each n ≥ 0.
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