|
==Hilbert-Pólya conjecture==
InAccording terms suggested byto 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> \sum = \sum_\rho F(\rho), \!</math>
is then formally equal to
would then have this interpretation: use the [[functional calculus]] of operators, supposed to apply to ''T'', to form
:<math> Tr(F(\widehat T )),\!</math>
which is (essentially) one side of the explicit formula.
and then take its [[Trace (linear algebra)|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} \int_{-\infty}^{\infty} e^{-aH(x,p)}\ dx\ dp,\!</math>
taking our operator <math>\scriptstyle F(\widehat T )\!</math> to be <math>e^{-\scriptstyle\widehat{H}}\!</math> valid when ''a'' is '' small '' and positive or pure imaginary.
Development of the explicit formulae for a wide class of L-functions was given by {{harvtxt|Weil|1952}}, 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.
| 