Explicit formulae for L-functions: Difference between revisions

Revision as of 02:14, 9 January 2009

Revision as of 02:15, 9 January 2009

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==Hilbert-Pólya conjecture==

According to the [[Hilbert-Pólya conjecture]], the complex zeroes ρ should be the [[eigenvalue]]s of some [[linear operator]] ''T''. AThe sum over the zeros of the explicit formula is then (at least formally) given by a trace:

:<math> \sum = \sum_\rho F(\rho), =Tr(F(\widehat T )).\!</math>

is then formally equal to

:<math> Tr(F(\widehat T )),\!</math>

which is (essentially) one side of the explicit formula.

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.