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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: ![]() [This is the simplest case of A. Weil's 1952 generalisation of the (number theoretical) explicit formula of Riemann and von Mangoldt. h is a complex-valued function of a real variable which satisfies certain conditions, and g is an integral transform of h. Further notes on this formula can be found here.] where the nontrivial zeros of the Riemann zeta function are denoted by ![]() D. Hejhal, The Selberg Trace Formula for PSL(2,R) - Volume I, p. 35 "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 Selberg zeta function; it is usually denoted by
Z(s)." [D. Hejhal, Duke Math. J. 1976, p.459]
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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