2016 Mini-Course by Chuck Newman - Statistical Mechanics and the Riemann Hypothesis

Topic: Statistical Mechanics and the Riemann Hypothesis
Date & Time: Tuesday, February 23, 2016 - 13:30 to Tuesday, March 22, 2016 - 15:30
Speaker: Charles Newman, Silver Professor of Mathematics, NYU/Affiliated Professor of Mathematics, NYU Shanghai
Location: Room 264, Geography Building, Zhongbei Campus, ECNU (中山北路校区,地理楼264室)

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The Mini-Course is sponsored by NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
© 2016 NYU Shanghai

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Every Tuesday, Feb. 23 - Mar. 22, 2016


We review a number of old results concerning certain statistical mechanics models and their possible connections to the Riemann Hypothesis.

A standard reformulation of the Riemann Hypothesis (RH) is: The (two-sided) Laplace transform of a certain specific function Ψ​ on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also Ψ​ is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function.

A (finite) Ising model is a specific type of probability measure P​ on the points S=(S1,...,SN)​ with each Sj = +1​ or -1​. The Lee-Yang theorem (of T. D. Lee and C. N. Yang) implies that for non-negative a1, ..., aN​, the Laplace transform of the induced probability distribution of a1 S1 + ... + aN SN​ has only pure imaginary zeros. There are also other models, where the variables are real-valued or vector-valued which have moment generating functions with only pure imaginary zeros.

An intriguing question is whether it's possible to find a sequence in N​ of models and generating functions  so that the limit as N → ∞​ of such distributions has density exactly C Ψ​. We'll discuss some of the cases where one can study the limiting distribution and some hints as to how one might try to find the "right'' choice.



Charles M. Newman, Silver Professor of Mathematics at the Courant Institute and Global Network Professor at NYU-New York and NYU-Shanghai, received B.S. degrees in Mathematics and in Physics from MIT and M.S. and Ph.D. degrees in Physics from Princeton. With 200+ published papers, mainly in probability and statistical physics, he has been a Sloan and Guggenheim fellow and is a member of the U.S. National Academy of Sciences, the American Academy of Arts and Sciences and the Brazilian Academy of Sciences.