Explicit formulae for L-functions: Difference between revisions

Revision as of 21:34, 6 September 2007

Revision as of 21:35, 6 September 2007

Line 15:

:<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 ''e''^{&nbsp;&minus;<math>\scriptstyle e^{-a\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 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.