Explicit formulae for L-functions: Difference between revisions

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In [[mathematics]], the '''explicit formulae for [[L-function]]s''' are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by {{harvtxt|Riemann|1859}} for the [[Riemann zeta function]]. 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]].

==Riemann's explicit formula==

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*<math>\Psi(t) = - \log( \pi ) + \operatorname{Re}(\psi(1/4 + it/2))</math>, where <math>\psi</math> is the [[digamma function]] Γ<big>''&prime;''</big>/Γ.

Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors. Once this is said, the formula comes from the fact that the Fourier transform is ana unitary operator, so that a scalar product in time ___domain is equal to the scalar product of the Fourier transforms in the frequency ___domain.

The terms in the formula arise in the following way.