Weil's criterion
Encyclopedia
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 function field
case. Here the inclusion of Artin L-function
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Weil's criterion is a criterion of André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
for the Generalized Riemann Hypothesis
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function
Generalized function
In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing physical phenomena such as point charges...
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 function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...
case. Here the inclusion of Artin L-function
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...
s, 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.