The triple reciprocity law for the twisted second moments of Dirichlet $L$-functions over function fields

2021 ◽  
pp. 1
Author(s):  
Goran Djanković ◽  
Dragan Đokić ◽  
Nikola Lelas
Keyword(s):  
Function Fields ◽  
Reciprocity Law ◽  
Second Moments

THE RECIPROCITY LAW FOR THE TWISTED SECOND MOMENT OF DIRICHLET -FUNCTIONS OVER RATIONAL FUNCTION FIELDS

2018 ◽  
Vol 98 (3) ◽  
pp. 383-388 ◽  
Author(s):  
GORAN DJANKOVIĆ
Keyword(s):  
Rational Function ◽  
Function Fields ◽  
Mean Square ◽  
Irreducible Polynomials ◽  
Second Moment ◽  
Reciprocity Law ◽  
Second Moments ◽  
The Mean

We prove the reciprocity law for the twisted second moments of Dirichlet $L$-functions over rational function fields, corresponding to two irreducible polynomials. This formula is the analogue of the formulas for Dirichlet $L$-functions over $\mathbb{Q}$ obtained by Conrey [‘The mean-square of Dirichlet $L$-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law for the twisted second moment of Dirichlet $L$-functions’, Forum Math. 23(6) (2011), 1323–1337].

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.

Multiple Dedekind–Rademacher sums in function fields

2014 ◽  
Vol 10 (05) ◽  
pp. 1291-1307 ◽  
Author(s):  
Abdelmejid Bayad ◽  
Yoshinori Hamahata
Keyword(s):  
Function Fields ◽  
Dedekind Sums ◽  
Reciprocity Law ◽  
Higher Dimensional

In the previous paper, we introduced the higher-dimensional Dedekind sums in function fields, and established the reciprocity law. In this paper, we generalize our higher-dimensional Dedekind sums and establish the reciprocity law and the Petersson–Knopp identity.

A rational reciprocity law over function fields

2016 ◽  
Vol 108 (3) ◽  
pp. 233-240
Author(s):  
Yoshinori Hamahata
Keyword(s):  
Function Fields ◽  
Reciprocity Law

Reciprocity Law for Compatible Systems of Abelian mod p Galois Representations

2005 ◽  
Vol 57 (6) ◽  
pp. 1215-1223 ◽  
Author(s):  
Chandrashekhar Khare
Keyword(s):  
Galois Representations ◽  
Function Fields ◽  
Arbitrary Dimension ◽  
Number Fields ◽  
Galois Groups ◽  
Reciprocity Law ◽  
Mod P

AbstractThe main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelianmod p representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer seems to require developments in transcendence theory in characteristic p.

scholarly journals Independently Controlling Stochastic Field Realization Magnitude and Phase Statistics for the Construction of Novel Partially Coherent Sources

Photonics ◽  
2021 ◽  
Vol 8 (2) ◽  
pp. 60
Author(s):  
Milo W. Hyde
Keyword(s):  
Coherent Beam ◽  
Optical Field ◽  
Stochastic Field ◽  
Partially Coherent ◽  
Second Moments ◽  
Coherent Beams ◽  
Gaussian Statistics ◽  
Simulated Field ◽  
Gaussian Moment ◽  
Partially Coherent Beams

In this paper, we present a method to independently control the field and irradiance statistics of a partially coherent beam. Prior techniques focus on generating optical field realizations whose ensemble-averaged autocorrelation matches a specified second-order field moment known as the cross-spectral density (CSD) function. Since optical field realizations are assumed to obey Gaussian statistics, these methods do not consider the irradiance moments, as they, by the Gaussian moment theorem, are completely determined by the field’s first and second moments. Our work, by including control over the irradiance statistics (in addition to the CSD function), expands existing synthesis approaches and allows for the design, modeling, and simulation of new partially coherent beams, whose underlying field realizations are not Gaussian distributed. We start with our model for a random optical field realization and then derive expressions relating the ensemble moments of our fields to those of the desired partially coherent beam. We describe in detail how to generate random optical field realizations with the proper statistics. We lastly generate two example partially coherent beams using our method and compare the simulated field and irradiance moments theory to validate our technique.

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