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 narrative of enquires often found in science. 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.

other condensed matterstatistical mechanicsstrongly correlated electronsgeneral relativity and quantum cosmologyHEP - theorymathematical physicsalgebraic geometrydifferential geometrydynamical systemsgroup theorynumber theoryquantum algebrarepresentation theory

Audience: researchers in the discipline

Number theory, Arithmetic and Algebraic Geometry, and Physics

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