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 aan 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