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In terms suggested by the [[Hilbert-Pólya conjecture]], one of the major heuristics underlying the [[Riemann hypothesis]] and its supposed connection with [[functional analysis]], the complex zeroes ρ should be closely linked to the [[eigenvalue]]s of some [[linear operator]] ''T''. A sum
:<math> \Sigma = \Sigma_{\rho} F(\rho) </math>
would then have this interpretation: use the [[functional calculus]] of operators, supposed to apply to ''T'', to form
:<math> F(\hat T ) </math>
and then take its [[trace of an operator|trace]]. In a formal sense, ignoring all the difficult points of [[mathematical analysis]] involved, this will be Σ. Therefore the existence of such 'trace formulae' for ''T'' means that the explicit formulae essentially encode the nature of ''T'', from the point of view of [[spectral theory]], at least as far as its eigenvalues ([[spectrum of an operator|spectrum]]) is concerned.
For the case the ''Spectrum '' is just the one belonging to a Hamiltonian H , the semiclassical approach can give a definition of the sum by means of an integral of the form:
:<math> \Sigma= \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty}dx e^{-aH(x,p)} </math>
taking our operator <math> F(\hat T ) </math> to be <math> e^{-a\hat H} valid when a is '' small '' and positive or pure imaginary.
Development of the explicit formulae for a wide class of L-functions took place in papers of [[André Weil]], who first extended the idea to [[local zeta-function]]s, and formulated a version of a [[generalized Riemann hypothesis]] in this setting, as a positivity statement for a [[generalized function]] on a [[topological group]]. More recent work by [[Alain Connes]] has gone much further into the functional-analytic background, providing a trace formula the validity of which is equivalent to such a generalized Riemann hypothesis.
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