Date: Wednesday, April 17, 2013
Location: 3866 East Hall (3:00 PM to 4:00 PM)
Title: Generalized prime numbers and integers
Abstract: The positive integers are a multiplicative semigroup generated by the prime numbers. In 1937, Beurling introduced "generalized integers," formed by taking the multiplicative semigroup generated by a sequence of positive real numbers called "generalized primes." What properties does a system of generalized integers have to satisfy in order for the classical Prime Number Theorem to hold? Can we construct a system of generalized integers for which the Prime Number Theorem holds but the Riemann Hypothesis fails? Answering these questions helps us understand which properties of the usual integers are used in the proof of the PNT and which properties will need to be accounted for in order to improve the PNT. I will give a survey of results of Beurling, Diamond, Malliavin, and others. Proofs will be sketched.
Speaker: Corey Everlove
Institution: UM
Event Organizer:
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