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- Weils_criterion abstract "In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of prime number theory, as they apply to Dirichlet L-functions, and other more general global L-functions. A single statement thus combines statements on the complex zeroes of all Dirichlet L-functions.Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (Artin-Hecke L-functions); and to the global function field case. Here the inclusion of Artin L-functions, in particular, implicates Artin's conjecture; so that the criterion involves a Generalized Riemann Hypothesis plus Artin Conjecture. The case of function fields, of curves over finite fields, is one in which the analogue of the Riemann Hypothesis is known, by Weil's classical work begun in 1940; and Weil also proved the analogue of the Artin Conjecture. Therefore in that setting, the criterion can be used to show the corresponding statement of positive-definiteness does hold.".
- Weils_criterion wikiPageID "5906027".
- Weils_criterion wikiPageLength "1732".
- Weils_criterion wikiPageOutDegree "13".
- Weils_criterion wikiPageRevisionID "645244793".
- Weils_criterion wikiPageWikiLink André_Weil.
- Weils_criterion wikiPageWikiLink Artin-Hecke_L-function.
- Weils_criterion wikiPageWikiLink Artin_L-function.
- Weils_criterion wikiPageWikiLink Artin_conjecture_(L-functions).
- Weils_criterion wikiPageWikiLink Category:Zeta_and_L-functions.
- Weils_criterion wikiPageWikiLink Dirichlet_L-function.
- Weils_criterion wikiPageWikiLink Explicit_formula_(L-function).
- Weils_criterion wikiPageWikiLink Explicit_formulae_(L-function).
- Weils_criterion wikiPageWikiLink Generalized_Riemann_hypothesis.
- Weils_criterion wikiPageWikiLink Generalized_function.
- Weils_criterion wikiPageWikiLink Global_L-function.
- Weils_criterion wikiPageWikiLink Global_field.
- Weils_criterion wikiPageWikiLink Global_function_field.
- Weils_criterion wikiPageWikiLink Mathematics.
- Weils_criterion wikiPageWikiLink Positive-definite_function.
- Weils_criterion wikiPageWikiLinkText "Weil's criterion".
- Weils_criterion hasPhotoCollection Weils_criterion.
- Weils_criterion subject Category:Zeta_and_L-functions.
- Weils_criterion hypernym Criterion.
- Weils_criterion comment "In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea was formulated first in a 1952 paper. It is based on the explicit formulae of prime number theory, as they apply to Dirichlet L-functions, and other more general global L-functions.".
- Weils_criterion label "Weil's criterion".
- Weils_criterion sameAs m.025tnmf.
- Weils_criterion sameAs Q7980177.
- Weils_criterion sameAs Q7980177.
- Weils_criterion wasDerivedFrom Weils_criterionoldid=645244793.
- Weils_criterion isPrimaryTopicOf Weils_criterion.