Colloquium Mathematics - Peter Veerman, University of Portland State

Title: Primes!

Abstract:

A non-technical review of some classical results in number theory. We identify some of the currents in number theory (analytic, algebraic, and ergodic), and informally discuss some results that had great impact on mathematical (and physical) thought. On the analytic side, we look at the prime number theorem (or PNT), which tells us how dense primes are in the natural numbers. While its proof starts with some combinatorial estimates, it turns out, very surprisingly, that the full proof makes essential use of complex analysis (the Cauchy integral formula). For that reason, this branch is now called analytic number theory. We will touch on algebraic number theory to extend the PNT to arithmetic progressions. This theorem gives the density of primes in sequences of the form {a+ib} where a and b are fixed integers. Time permitting we will very briefly mention ergodic theory.