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From quantum systems to \(L\)-functions: pair correlation statistics and beyond. (English) Zbl 1351.11049

Nash, John Forbes jun. (ed.) et al., Open problems in mathematics. Cham: Springer (ISBN 978-3-319-32160-8/hbk; 978-3-319-32162-2/ebook). 123-171 (2016).
Summary: The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through H. L. Montgomery’s work [in: Analytic Number Theory, Proc. Sympos. Pure Math. 24, St. Louis Univ. Missouri 1972, 181–193 (1973; Zbl 0268.10023)] on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.
For the entire collection see [Zbl 1351.00027].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M50 Relations with random matrices
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices

Citations:

Zbl 0268.10023
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