Elementary proofs of the abstract prime number theorem for algebraic function fields

Abstract

Elementary proofs of the abstract prime number theorem of the form Λ ( m ) = q m + O ( q m m − 1 ) \Lambda (m) = {q^m} + O({q^m}{m^{ - 1}}) for algebraic function fields are given. The proofs use a refinement of a tauberian theorem of Bombieri.