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K-State Today

January 26, 2016

Mathematics Colloquium lecture Jan. 26

By Reta McDermott

Tristan Frieberg, University of Waterloo, will present "Prime Gaps: Small, Medium and Large" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Tuesday, Jan. 26, in 122 Cardwell Hall.

The abstract for the lecture is: It's an oft-repeated story that Gauss, at an early age, ''directed [his] attention to the decreasing frequency of primes''. He observed that, ''behind all its fluctuations'', the nth prime gap, d_n=p_{n+1}−p_n (where p_n is the nth prime), is very close to log n {\em on average}. This is affirmed by the prime number theorem, arguably the pinnacle of 19th century number theory, for whose proof a program was laid out by Riemann in his seminal 1859 memoir, and completed by Hadamard and de la Vallee-Poussin in 1896.

As to those ''fluctuations'' — e.g. how often is d_n/log n less than 1/2, or greater than 100? — over a century later we have very little more than conjecture to go on. However, by combining the celebrated breakthrough work of Maynard and Tao, originally developed to produce bounded gaps between primes, with a construction of Erd{\H o}s and Rankin for producing {\em long} gaps between consecutive primes, we are finally able to make some (very modest) headway on questions concerning the finer points of the distribution of primes.

This includes joint work with William Banks and James Maynard, and joint work with Roger Baker.

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