Speaker: Hugo Chapdelaine, Universite Laval https://www.mat.ulaval.ca/hchapdelaine/hugo-chapdelaine
Title: The real analytic Eisenstein series associated to the classical modular curve and some of its applications.
Date: Wednesday, October 31, 2018
Time: 2:30-3:30 p.m.
Place:  HP 4351 (MacPhail Room) School of Mathematics & Statistics Carleton University

Abstract: Firstly, I shall recall some basic notions in hyperbolic geometry using the Poincare upper half-plane model. Secondly, I will introduce the main protagonist of the talk, namely a family of real analytic Eisenstein series E(z,s) associated to the classical modular group SL_2(Z). Here z is a variable lying in the Poincare upper half-plane, while the variable s, thought of as the family parameter, belongs to a suitable right half-plane of the complex plane. We shall then proceed by stating some remarkable properties of this family, in particular its relationship with the hyperbolic Laplacian and its close connection with the classical Riemann zeta function. As a first application, we shall explain how this family, when specialized to the vertical line Re(s)=1/2, gives a “physical incarnation” of the continuous spectrum of the hyperbolic Laplacian. As a final application, we shall explain in some detail how one can reprove the Prime Number Theorem from the key observation that E(z,s) does not vanish identically, as a function of z, when s is specialized to any number on the vertical line Re(s)=1/2.  The talk will be kept at an elementary level but some basic knowledge on the classical Fourier transform on the real line and on the Fourier series of periodic functions will turn out to be essential.