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In [[mathematics]], the '''explicit formulae for [[L-function]]s''' relate sums taken over the complex number zeroes of a given L-function to sums over prime powers. The first case was found by Riemann 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]].
==Explicit formula for the Riemann zeta function==
The explicit formul states
:<math>\Phi(1)+\Phi(0)-\sum_\rho\Phi(\rho)
=
\sum_{p,m}\frac{\log(p)}{p^{m/2}} (F(\log(p^m)+F(-\log(p^m)) -\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi(t)\Psi(t)dt</math>
==Hilbert-Pólya conjecture==
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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