The Riemann Hypothesis for Function Fields

1986 ◽  
pp. 28-42
Author(s):  
Michael D. Fried ◽  
Moshe Jarden
Keyword(s):  
Riemann Hypothesis ◽  
Function Fields

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

The Riemann Hypothesis for Function Fields

2009 ◽  
Author(s):  
Machiel van Frankenhuijsen
Keyword(s):  
Riemann Hypothesis ◽  
Function Fields

scholarly journals The distribution of the residues of a quartic polynomial

1967 ◽  
Vol 8 (2) ◽  
pp. 67-88 ◽  
Author(s):  
K. McCann ◽  
K. S. Williams
Keyword(s):  
Finite Field ◽  
Finite Number ◽  
Galois Group ◽  
Riemann Hypothesis ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Closure ◽  
Quartic Polynomial ◽  
Algebraic Function Fields ◽  
Image Position

Let f(x) denote a polynomial of degree d defined over a finite field k with q = pnelements. B. J. Birch and H. P. F. Swinnerton-Dyer [1] have estimated the number N(f) of distinct values of y in k for which at least one of the roots ofis in k. They prove, using A. Weil's deep results [12] (that is, results depending on the Riemann hypothesis for algebraic function fields over a finite field) on the number of points on a finite number of curves, thatwhere λ is a certain constant and the constant implied by the O-symbol depends only on d. In fact, if G(f) denotes the Galois group of the equation (1.1) over k(y) and G+(f) its Galois group over k+(y), where k+ is the algebraic closure of k, then it is shown that λ depends only on G(f), G+(f) and d. It is pointed out that “in general”

On the Number of Solutions of Some General Types of Equations in a Finite Field

1952 ◽  
Vol 4 ◽  
pp. 343-351 ◽  
Author(s):  
Olin B. Faircloth
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Riemann Hypothesis ◽  
Quadratic Forms ◽  
Function Fields ◽  
Commutative Rings ◽  
Number Of Solutions ◽  
Fermat's Last Theorem ◽  
Conditional Equation ◽  
And Algebra

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.

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