Sur les Formules Explicites I: analyse invariante

Abstract

Weil has generalized the Riemann-von Mangoldt explicit formula linking the prime numbers with the zeros of the zeta function to the set-up of a general algebraic number field K and Dirichlet-Hecke L-function, revealing in the process the role played by the completions (finite and infinite) of K. We show how the local terms of these explicit formulae are explained by the dilaton invariant ``conductor operator'' log(|x|) + log(|y|). We also check Weil's positivity criterion under a support condition.

6 pages. Preprint version of the published note. Mainly in French with an abridged version in English

Cited by in corpus (10)

• On a representation of the idele class group related to primes and zeros of L-functions
• On Fourier and Zeta(s)
• A lower bound in an approximation problem involving the zeros of the Riemann zeta function
• Sur certains espaces de Hilbert de fonctions entieres, lies a la transformation de Fourier et aux fonctions L de Dirichlet et de Riemann
• Des equations de Dirac et de Schrodinger pour la transformation de Fourier
• Li Coefficients for Automorphic L-Functions
• Two complete and minimal systems associated with the zeros of the Riemann zeta function
• The Lerch zeta function and the Heisenberg group
• Explicit formulas for the spectral side of the trace formula of SL(2)
• Quaternionic Gamma functions and their logarithmic derivatives as spectral functions