Friday, February 25, 4-5 pm, MW 154

In this talk Shifan will introduce the classical theory of Dirichlet characters and Dirichlet L-functions. They were first introduced by Dirichlet to prove his famous theorem on primes in arithmetic progressions. And it turns out the key of his proof lie on the non-vanishing of those L-functions at 1. After Riemann’s epoch memoir in 1859, people realized that there is a deep connection between the distribution of primes and the zero-free regions of Dirichlet L-functions (Riemann zeta function included as a special case). A wider zero-free region was soon discoverd, for complex primitive characters, which led to a quantitive version of Dirichlet’s theorem. Yet the same type of zero-free region does not hold for real primitive character, due to the so called “Siegel zeros”, whose existence is still unknown. Eliminating those zeros will be the first step to the generalized Riemann hypothesis.