In [[mathematics]], the '''explicit formulae for [[L-function]]s''' arerelate a class of summation formulae, expressing sums taken over the complex number zeroes of a given L-function,typicallyto insums termsover ofprime quantitiespowers. studiedbyThe [[numberfirst theory]]case bywas usefound ofby theRiemann 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 an algebraic 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 ρ should be closely linked to the [[eigenvalue]]s of some [[linear operator]] ''T''. A sum
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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 tookwas placegiven in papers of [[Andréby {{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.
