Generalised Riemann Hypothesis and Brownian Motion

ABSTRACT
If Number Theory is arguably one of the most fascinating subjects in Mathematics, Theoretical Physics adds to it the standard of clarity, beauty and deepness which have helped us to shape our understanding of the laws of Nature: together, these two subjects present a fascinating story worth telling, one of those vital, wonderful and superb narratives of enquires rarely found in human history. From this point of view, the seminar presents the main features of the Riemann Hypothesis and discusses its generalisation to an infinite class of complex functions, the so-called Dirichlet L-functions, regarded as quantum partition functions on the prime numbers. The position of the infinite number of zeros of all the Dirichlet L-functions along the axis with real part equal to 1/2 finds a very natural explanation in terms of one of the most basic phenomena in Statistical Physics, alias the Brownian motion. We present the probabilistic arguments which lead to this conclusion and we also discuss a battery of highly non-trivial tests which support with an extremely high confidence the validity of this result.
BIO
Giuseppe Mussardo is Professor of Theoretical Physics at SISSA in Trieste, where he founded and directed the Statistical Physics group for several years. He is the Editorial Director of the "Journal of Statistical Physics and Applications" (JSTAT) and author of various scientific monographs, such as "Statistical Field Theory", published by Oxford University Press. He has promoted research in quantum integrable models, where for example he calculated for the first time exactly the correlation functions of the Ising model in the magnetic field, the exact S-matrix of the Yang-Lee model and Toda Field Theories, the correlation functions of the Lieb-Liniger model for cold atoms, and open the study of Truncated Hilbert Space Approach and quantum field theories out of equilibrium. Among his recent research fields there is the relation between Number Theory and Quantum Statistical Physics: in this topic he has proposed and determined exactly a quantum potential having prime numbers as the all and only eigenvalues, and promoted the study of the Riemann conjecture with the methods of statistical mechanics.

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