Heading: L-Functions Of Real Quadratic Characters

Contents

  1. Definition
    1. Dirichlet Series & Euler Product
    2. Functional Equation
    3. Fundamental Discriminants
    4. Kronecker Symbol
  2. Moments
    1. RMT Conjectures
  3. Applications

Definition

These L-functions are a subset of the Dirichlet L-functions having character \chi_d(n) given by the Kronecker Symbol \left(\frac{d}{n}\right)

Dirichlet Series & Euler Product

These L-functions have Dirichlet series and Euler product

L(s,\chi_d)=\sum_{n=1}^\infty \frac{\chi_d(n)}{n^s} =\prod_{p}\left(1-\frac{\chi_d(p)}{p^s}\right)^{-1},
which are absolutely convergent for \Re{(s)}>1.

\chi_d(n) are called the real quadratic characters, and \chi_d(n) is a primitive character (hence L(s,\chi_d) is a primitive) for fundamental discriminants d.

Functional Equation

The completed L-function is dependent on the sign of d, and is given by

\xi(s,\chi_{d})=\left(\frac{|d|}{\pi}\right)^{\frac{s+a}{2}}\Gamma\left(\frac{s+a}{2}\right)L(s,\chi_d),

where

a= \begin{cases} 0 & \text{if }d>0,\\ 1 & \text{if }d<0. \end{cases}

\xi(s,\chi_d) is entire and satisfies the functional equation \xi(s,\chi_d)=\xi(1-s,\chi_d).

Fundamental Discriminants

d is fundamental discriminant if:

Kronecker Symbol

The Kronecker Symbol, (\frac{d}{n}), is a completely multiplicative extension of the Legendre and Jacobi Symbols to all integers.

Moments

RMT Conjectures

Applications

L-functions/LFunctionsOfRealQuadraticCharacters (last edited 2009-03-01 00:55:51 by localhost)