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Thursday, July 7, 2016

**Abstract:** In classical prime number theory several asymptotic relations are considered to be "equivalent" with the prime number theorem, meaning that they could be deduced by simple real variable arguments. In the context of Beurling numbers, however, this is no longer the case: sometimes extra hypotheses have to be imposed to show the equivalence between different asymptotic relations. We will present some recent results on the subject. In contrast to the earlier work of Diamond and Zhang, who used strictly elementary methods, our approach will make extensive use of the zeta function of the prime number system and can thus no longer be regarded as elementary. The talk is based on collaborative work with Jasson Vindas.