In a textbook of 1946, Erwin Schrödinger was working out the behaviour of an ideal quantum gas, and needed a number. He noted in passing that the series he had arrived at were, apart from simple factors, the Riemann zeta function – the function that governs how the prime numbers are distributed – "which can be found in tables". No comment. Why would there be a comment? He needed a number, and the number was in a table.
This archive is what happens if you take that parenthesis seriously for twenty-five years. It attempts to document, in as much detail as possible, the research which in some way links number theory and physics: number theory being applied by physicists to solve physical problems, and, perhaps more unexpectedly, techniques developed by physicists being applied to problems in number theory.
Where number theory and physics meet directly. Each section collects papers, preprints, excerpts and commentary.
quantum mechanics statistical mechanics p-adic and adelic physics Selberg trace formula string theory and quantum cosmology scattering dynamical and spectral zeta functions trace formulae and explicit formulae 1/f noise and signal processing supersymmetry QCD renormalisation symmetry breaking and phase transitions quantum fields integer partitions time biologically-inspired methods for finding primes dynamical systems entropy specific zeta values
Sections that grew up around the central question. Some are not physics at all – probability and statistics, for instance, were developed to describe the changing physical world, and their applicability to something as unchanging as the primes is itself part of the puzzle.
probability and statistics noncommutative geometry random matrices Fourier theory fractal geometry Bernoulli numbers Farey sequences Beurling g-primes Golden mean logic, languages, information, etc.
directory of zeta functions directory of L-functions conferences miscellaneous reception and correspondence quotations
FAQ and resources Riemann's original paper proposed proofs reformulations Critical Strip Explorer
Assembled by Matthew Watkins and hosted by the
University of Exeter since 2000. Contributions are welcome; the archive has always
erred towards inclusion, and material lying outside the accepted literature is
flagged where it appears rather than excluded.
Communications in Number
Theory and Physics, a journal founded in 2007, covers much of this ground.
The animation above is by Raymond Manzoni –
what you are watching.