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Dirichlet theta function

Contents

Idea

Given a Dirichlet character

χ:(/N) × × \chi \;\colon\; (\mathbb{Z}/N\mathbb{Z})^\times \longrightarrow \mathbb{C}^\times

it is called even if χ(1)=1\chi(-1) = 1 and odd if χ(1)=1\chi(-1) = -1.

The theta function associated with χ\chi is the function on +\mathbb{R}_+ defined by

θ χ(τ){nχ(n)exp(πn 2t/N) ifχeven nnχ(n)exp(πn 2t/N) ifχodd \theta_\chi(\tau) \coloneqq \left\{ \array{ \underset{n \in \mathbb{Z}}{\sum} \chi(n) \exp(-\pi n^2 t/N) & if\; \chi\; even \\ \underset{n \in \mathbb{Z}}{\sum} n \chi(n) \exp(-\pi n^2 t/N) & if\; \chi\; odd } \right.

where NN is the conductor of χ\chi.

Properties

Relation to Jacobi theta function

For trivial χ\chi the Dirichlet theta reproduces the Jacobi theta function.

Mellin transform and relation to Dirichlet L-function

The Mellin transform of this is the completed Dirichlet L-function associated with χ\chi (e.g. Continuations, p. 8)

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

  • section 3 of Continuations and functional equations (pdf)

Last revised on September 11, 2014 at 01:57:24. See the history of this page for a list of all contributions to it.