# Riemann Hypothesis and Mathematical Physics

### Riemann Hypothesis and Mathematical Physics

Series: Mathematics Colloquium
Location: Math 501
Presenter: Charles M. Newman, Courant Institute

Abstract: In both analytic number theory (the Riemann Hypothesis) and
mathematical physics (Ising models and Euclidean field theories) the
following complex analysis issue arises. For \rho a finite positive
measure on the real line R, let H(z; \rho, \lambda) denote the Fourier
transform of \exp{\lambda u^2} d\rho (u), i.e., the integral over R of
\exp{izu + \lambda u^2} d\rho (u) extended from real to complex z, for
those \lambda (including all \lambda < 0) where this is possible. The
issue is to determine for various \rho's those \lambda's for which all
zeros of H in the complex plane are real. We will discuss some old and new