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Dirichlet L-function
Redirected from "Dirichlet L-functions".
Contents
Context
Theta functions
Arithmetic geometry
number theory
number
natural number , integer number , rational number , real number , irrational number , complex number , quaternion , octonion , adic number , cardinal number , ordinal number , surreal number
arithmetic
arithmetic geometry , function field analogy
Arakelov geometry
Contents
Idea
The Dirichlet L-functions L χ L_\chi are a kind of L-function induced by Dirichlet characters χ \chi
L χ ( s ) ≔ ∑ n = 1 ∞ χ ( 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 ↦ ∫ 𝕀 χ ( x ) f ( x ) | x | s
s \mapsto \int_{\mathbb{I}} \chi(x) \;f(x)\; {\vert x\vert}^s
for suitable Schwartz functions f 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 n = 1 ) the generalization to automorphic forms for n ≥ 1 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.
context/function field analogy theta function θ \theta zeta function ζ \zeta (= Mellin transform of θ ( 0 , − ) \theta(0,-) )L-function L z L_{\mathbf{z}} (= Mellin transform of θ ( z , − ) \theta(\mathbf{z},-) )eta function η \eta special values of L-functions physics /2d CFT partition 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 s − 1 θ ( 0 , τ ) d τ \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 z \mathbf{z} : L z ( s ) ≔ Tr reg ( 1 ( D z ) 2 ) s = ∫ 0 ∞ τ s − 1 θ ( z , τ ) d τ 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 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 pv L 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 pv η 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 − 1 2 L z ′ ( 0 ) = Z H = 1 2 ln det 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 operator zeta function of an elliptic differential operator eta function of a self-adjoint operator functional determinant , analytic torsion
complex analytic geometry section 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 ℂ g → J ( Σ τ ) \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 z L_{\mathbf{z}} for z = 0 \mathbf{z} = 0 the trivial Galois representation )Artin L-function L z L_{\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 ℚ \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 z L_{\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 L 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 )
Last revised on December 13, 2020 at 17:15:10.
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