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Dirichlet L-function

Contents

Contents

Idea

The Dirichlet L-functions L χL_\chi are a kind of L-function induced by Dirichlet characters χ\chi

L χ(s)n=1χ(n)n s. L_\chi(s) \coloneqq \underoverset{n=1}{\infty}{\sum} \chi(n)n^{-s} \,.

(e.g. Goldfeld-Hundley 11 (2.2.1), Continuations).

The completion of this by a Gamma function factor (and a power of π\pi and of the conductor) is the Mellin transform of the Dirichlet theta function (e.g. Continuations, p. 8).

By Artin reciprocity Dirichlet L-function are equal to suitable Artin L-functions induced by 1-dimensional Galois representations.

In terms of the adelic integral representation of L-functions via Iwasawa-Tate theory, given a Dirichlet character χ\chi then the corresponding Dirichlet L-functions are simply the adelic integrals (e.g. Garrett 11, section 2.2)

s 𝕀χ(x)f(x)|x| s s \mapsto \int_{\mathbb{I}} \chi(x) \;f(x)\; {\vert x\vert}^s

for suitable Schwartz functions ff on the idele group.

Hence Dirichlet L-functions are Mellin transforms of suitable theta functions (“theta kernels”), see. e.g. (Stopple, p. 3).

Where Dirichlet characters (see there) are essentially automorphic forms for the idele group (hence for n=1n = 1) the generalization to automorphic forms for n1n \geq 1 is the concept of automorphic L-function.

The differential geometric analog of a Dirichlet L-function is the eta function (see there for more) of a differential operator.

context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of θ(z,)\theta(\mathbf{z},-))eta function η\etaspecial values of L-functions
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued trace of Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy 12L z (0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\mathbf{z}, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number \cdot regulator
arithmetic geometry for \mathbb{Q}Jacobi theta function (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=χ\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\mathbf{z} , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

  • E. Kowalski, section 1.3 of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)

  • section 3 of Continuations and functional equations (pdf)

  • Wikipedia, Dirichlet L-function

  • Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

  • Paul Garrett, section 1.6 Iwasawa-Tate on ζ-functions and L-functions

    (pdf)

  • Jeffrey Stopple, Theta and LL-function splittings, Acta Arithmetica LXXII.2 (1995) (pdf)

Analogs of Dirichlet L-functions in chromatic homotopy theory are constructed in

  • Ningchuan Zhang, Analogs of Dirichlet L-functions in chromatic homotopy theory, (arXiv:1910.14582)

Last revised on December 13, 2020 at 12:15:10. See the history of this page for a list of all contributions to it.