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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
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A. Weil, "Sur les formules explicites de la théorie des nombres, Izvestia Akad. Nauk S.S.S.R., Ser. Math. 36 (1972) 3-18; Collected Papers III, 249-264
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A. Weil, "Sur les 'formules explicites' de la théorie des nombres premiers", Comm. Lund (vol. dédié a Marcel Riesz) (1952) 252–265; Collected Papers II
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Weil returned to this idea in a 1972 paper, showing how the formulation extended to a larger class of L-functions (
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to be true. It takes the form of an equivalent statement, to the effect that a certain
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55:. A single statement thus combines statements on the complex zeroes of
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Weil's idea was formulated first in a 1952 paper. It is based on the
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corresponding statement of positive-definiteness does hold.
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47:of prime number theory, as they apply to
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30:Generalized Riemann hypothesis
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70:case. Here the inclusion of
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51:, and other more general
64:Artin-Hecke L-functions
59:Dirichlet L-functions.
68:global function field
49:Dirichlet L-functions
108:Zeta and L-functions
34:generalized function
76:Artin's conjecture
53:global L-functions
24:is a criterion of
72:Artin L-functions
45:explicit formulae
38:positive definite
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22:Weil's criterion
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18:mathematics
86:References
26:André Weil
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28:for the
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16:In
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