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Tuesday, July 5, 2016

**Abstract:** This talk is a survey of classical and recent results on the asymptotic distribution of generalized prime numbers. The goal of Beurling's generalized prime number theory is to replace the ordinary prime numbers by a rather arbitrary non-decreasing sequence of positive real numbers (generalized primes), consider then the multiplicative semigroup generated by it (generalized integers), and then establish relations between asymptotic properties of the counting functions of the generalized primes and integers. We will discuss various conditions that ensure the validity of the prime number theorem in this context.