Description
The Riemann Hypothesis is one of the most famous open problems in mathematics; indeed, there is a prize of one million dollars for its solution.
The hypothesis concerns the locations of the (non-trivial) zeros of a special function, called the Riemann zeta function. Riemann realised in 1859 that these zeros encode the distribution of the prime numbers; and if these zeros were all to lie on a certain line in the complex plane, then many conjectured properties of the primes would follow.
However, in spite of almost 150 years of work, it is still unknown whether the hypothesis is true or not. One of the most remarkable recent developments in the story came in 1972, with a chance conversation over tea at the Institute for Advanced Study, Princeton, between Freeman Dyson, a physicist, and Hugh Montgomery, a number theorist. They realised that there was a link between Riemann's zeta function and the behaviour of the energy levels of large atomic nuclei, when these nuclei are modelled using large random matrices.
This unexpected connection has raised hopes that a proof of Riemann's hypothesis might be a step closer, and new developments are being reported to this day.
A project on this topic could begin by describing the Riemann zeta function, the Riemann Hypothesis, and their connection with the primes, before explaining Dyson and Montgomery's observation and the link with random matrices. Further directions might include a look at other sorts of zeta functions, and perhaps connections with quantum chaos.
Prerequisites
Complex analysis; some idea about quantum mechanics would be useful but not essential. This topic should appeal particularly to those with an interest in the exchange of ideas between pure mathematics and physics.Resources
John Derbyshire's book explains the history, and some of the mathematics, of the Riemann Hypothesis in a very accessible way.
The Clay Mathematics Institute description of the Riemann hypothesis.
A popular article describing the links between Riemann and random matrices is here, while this link gives a more detailed account of the story.
Collections of references on links between Riemann's zeta function and random matrices, some quite advanced, can be found here and here.
A survey from the point of view of computability is given by Odlyzko, while Diaconis gives a short description of what a random matrix is.See also this entry on mathworld.
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