Zeta Values as Moments: A Random Matrix Theory
par
Amphithéâtre Léon Motchane
IHES
The values of the Riemann zeta-function $\zeta\left(\alpha+\beta n\right)$ with $\alpha>1$, $\beta>0$ and $n$ a positive integer can be viewed as moments of a probability distribution. In my talk I will construct the corresponding random matrix model and determine its asymptotic spectral curve, which exhibits an unusual feature, the complete asymptotics of its partition function and uniform approximations to its correlation functions using a discrete version of Riemann-Hilbert problem with Airy and Bessel asymptotics. This relates a problem in number theory to ideas of Vershik and Okounkov in the representation theory of the symmetric group.
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Danylo Radchenko