Explicit formulae for L-functions: Difference between revisions

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In [[mathematics]], the '''explicit formulae for [[L-function]]s''' are a class of summation formulae, expressing sums taken over the complex number zeroes of a given L-function, typically in terms of quantities studied by [[number theory]] by use of the theory of [[special function]]s. The first case known was for the [[Riemann zeta function]], where sums over its complex zeroes are identified with other sums over [[prime number]]s. Such explicit formulae have been applied also to questions on bounding the [[discriminant of a number field]], and the [[conductor of a number field]].

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 ρρ should be closely linked to the [[eigenvalue]]s of some [[linear operator]] ''T''. A sum

:ΣΣ = ΣΣ ''F''(ρρ)

would then have this interpretation: use the [[functional calculus]] of operators, supposed to apply to ''T'', to form

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:''F''(''T'')

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 ΣΣ. 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.

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.