nLab Dirichlet L-function

Idea

The Dirichlet L-functions $L_\chi$ are a kind of L-function induced by Dirichlet characters $\chi$

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

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

The completion of this by a Gamma function factor (and a power of $\pi$ 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 $\chi$ then the corresponding Dirichlet L-functions are simply the adelic integrals (e.g. Garrett 11, section 2.2)

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

for suitable Schwartz functions $f$ 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 $n = 1$) the generalization to automorphic forms for $n \geq 1$ 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.

Related concepts

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, 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 ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , 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 $L$-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)