The Riemann Hypothesis and Stein's Method

Qi-Man Shao  (The Chinese University of Hong Kong)

14:00 pm - 15:00 pm, May 4th, 2016    Science Building A1510
（闵行校区数学楼 401 同步视频直播）

Abstract:

The Rieman hypothesis is a well-known open question and there are many equivalent statements. One equivalent conjecture is that the moment generating function of a special probability density function, say $\Psi$, has pure imaginary zeros. So the conjecture is reduced to find a sequence of random variables whose moment generating functions have pure imaginary zeros and their limiting probability density function is $\Psi$. It was proved by Lee-Yang (1952) that the moment generating function of Ising models has pure imaginary zeros. Therefore, if one can find a sequence of Ising models so that the limiting probability density function is $\Psi$, then the Riemann Hypothesis holds.

In this talk we shall use Stein's method to give a concrete approach to identify the limiting distribution for any given sequence of Ising models. The problem can be reduced to calculate conditional expectations and conditional variances.