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"It is easy to understand why Selberg studied trace formulas so intensively: they bear a very striking resemblance to the so-called explicit formulas of prime number theory. Briefly stated, one has: [ where the nontrivial zeros of the Riemann zeta function are denoted by ^{"}D. Hejhal, "Selberg noticed this similarity in 1950-51 and was quickly led to
a deeper study of trace formulas. Among other things, Selberg found
that there is a zeta function which corresponds to [his trace formula]
in the same way that [the Riemann zeta function] corresponds to
[Weil's explicit formula]. This zeta function is nowadays referred to
as the One possible starting point is the Poisson summation formula. This
arises initially as an identity in Fourier analysis, but can be
interpreted as a simple 'trace formula' on the torus, as P. Perry has
explained. Selberg generalised this, producing a noncommutative
analogue of the PSF, which can be interpreted as a trace formula on
higher-genus compact Riemannian surfaces. He noted the similarity
with a certain form of Weil's explicit formula wherein the nontrivial
zeta zeros correspond to eigenvalues. This provides some circumstantial
evidence for the Hilbert-Pólya conjecture, and suggests that the
explicit formulae of number theory might be trace formulae in some
yet-to-be-discovered context.
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