nLab Riemann zeta function

Idea

The Riemann zeta function is the archetypical example of a zeta function, defined by the formula

$$\zeta \colon s \mapsto \underoverset{n = 1}{\infty}{\sum} \frac{1}{n^s} \,.$$

From the point of view of arithmetic geometry and the function field analogy, the Riemann zeta function is the basic case “over F1” of a tower of zeta functions for arithmetic curves given by more general number fields – the Dedekind zeta functions – and over function fields – the Weil zeta function – and for complex curves – the Selberg zeta function of a Riemann surface – and in another direction for higher dimensional arithmetic schemes – the arithmetic zeta functions.

Properties

Special values

Some of special values of the Riemann zeta function found (for the non-trivial region of non-positive integers) by Leonhard Euler in 1734 and 1749 are

n	-5	-3	-1	0	2	4	6	8
$\zeta(n)$	$-\frac{1}{252}$	$\frac{1}{120}$	$-\frac{1}{12}$	$-\frac{1}{2}$	$\frac{\pi^2}{6}$	$\frac{\pi^4}{90}$	$\frac{\pi^6}{945}$	$\frac{\pi^8}{9450}$

where for instance the value $\zeta(-1) = -\frac{1}{12}$ turns out to be the Euler characteristic of the moduli stack of complex elliptic curves and as such controls much of string theory.

The completed zeta function

The following slight variant of the actual Riemann zeta function typically exhibits its special properties more explicitly.

Definition

The completed Riemann zeta function is

$$\hat \zeta(s) \coloneqq \pi^{-s/2}\Gamma(s/2)\zeta(s) \,,$$

where $\Gamma(-)$ denotes the Gamma function.

Proposition

The completed Riemann zeta function, def. , is the adelic integral

$$\hat \zeta(s) = \int_{\mathbb{A}_{\mathbb{Q}}^\times } \exp(-S(x)) \; {\vert x\vert}^s \; d \mu_{\mathbb{A}_{\mathbb{Q}}^\times}(x)$$

where

• $\exp(-S(-))\colon \mathbb{A}_{\mathbb{Q}}^\times$ denotes the function which sends an idele $x \in \mathbb{A}_{\mathbb{Q}}^\times$ with canonical components $x = (x_\infty, x_2, \cdots, x_p, \cdots)$

  to the product

  $$\exp(-S(x)) \coloneqq \exp(-\pi x_\infty^2)\prod_{p \; prime} \chi_{\mathbb{Z}_p}(x_p) \,,$$

  where $\chi_{\mathbb{Z}_p}$ denotes the characteristic function of the p-adic integers inside the ring of adeles;

  (Goldfeld-Hundley 11, def. 2.2.5).

• the measure is essentially the Haar measure on the idele group

  $$d \mu_{\mathbb{A}_{\mathbb{Q}}^\times} \coloneqq \underset{p \leq \infty}{\prod} d^\times x_p$$
  $$d^\times x_p \coloneqq \left\{ \array{ \frac{d x_\infty}{{\vert x_\infty\vert}_\infty} & if \; p = \infty \\ \frac{1}{1-p^{-1}}\frac{d x_p}{{\vert x_p\vert}_p} & if\; p \; finite \; prime } \right.$$

  (Goldfeld-Hundley 11, def. 2.2.3)

(reviewed e.g. in Fesenko 08 0.1, Garrett 11, section 1, Goldfeld-Hundley 11 (2.2.6)).

Relation to the Jacobi theta function

Proposition

The completed zeta function, def. , is the Mellin transform of the Jacobi theta function

$$\theta(x)\coloneqq \underset{n \in \mathbb{Z}}{\sum} \exp(- \pi n ^2 x)$$

in that

$$\begin{aligned} \hat \zeta(s) &= \int_0^\infty (\theta(x^2)-1) x^s \frac{d x}{x} \\ & \stackrel{\tau := x^2}{=} \frac{1}{2} \int_0^\infty (\theta(\tau)-1) \tau^{s/2} \frac{d \tau}{\tau} \end{aligned} \,.$$

e.g. (Fesenko 08, section 0.1, Kowalski, example 2.2.5)

In terms of idelic integral expression for the complete zeta-function of prop. , this comes out as follows:

We compute the integral $\int_{\mathbb{A}^\times} \exp(-S(\alpha))) |\alpha|^s d\mu_{\mathbb{A}^\times}(\alpha)$ – as in (Goldfeld-Hundley 11, pages 47-50) and the remarks by Ivan Fesenko in (Goldfeld-Hundley 11, pages 51-51).

We decompose by the strong approximation theorem for ideles the integration domain into the idele class group

$$\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times = (0,\infty) \times \underset{p}{\prod} \mathbb{Z}_p^\times$$

and a factor of the non-zero rational numbers: so we write

$$\alpha=x n$$

where $x$ runs through representatives of $\mathbb{A}^\times/\mathbb{Q}^\times$ and can be chosen as ideles $x=(x_2,x_3,...x_\infty)$ with non-archimedean coordinates being units in $\mathbb{Z}_p$ and $x_\infty$ a positive real number, and $n$ is a non-zero rational number.

This way the inner integration is $\int_{\mathbb{Q}^\times} \exp(-S(x n)) d n$. Due to the definition of $\exp(-S(-))$ in prop. , the integrand here is supported on elements $x n\in \mathbb{Z}_p$ for all $p$, and since $x_p\in \mathbb{Z}_p^\times$ we deduce $n\in \mathbb{Z}_p$ for all $p$. Since $\mathbb{Q}$ intersected with all $\mathbb{Z}_p$ is $\mathbb{Z}$ (by the arithmetic fracture square), we have

$$\begin{aligned} \int_{\mathbb{Q}^\times}\exp(-S(x n)) d n & = \sum_{n\in \mathbb{Z}^\times} exp(-\pi n^2 x_\infty^2) \\ & = \theta(x_\infty^2)-1 \end{aligned}$$

Therefore the full integral becomes $\int_{\mathbb{R}_{\gt 0}}( \theta(y^2)-1) y^s \frac{d y}{y}$ with $y=x_\infty$.

Functional equation

The adelic integral representation of prop. directly implies the functional equation

$$\hat \zeta(1-s) = \hat \zeta(s)$$

of the completed zeta function from the functional equation of the theta function

$$\theta(x^2) x = \theta(x^{-2})$$

which in turn follows from the Poisson summation formula (see at Jacobi theta function – Functional equation).

In terms of the adelic integral expression, the functional equation of the theta function (and of the zeta integral) corresponds to the analytic duality furnished by Fourier transform on the adelic spaces and its subspaces. (due to Tate 50, reviewed for instance in Fesenko 08 0.1, Garrett 11, section 1.9, Goldfeld-Hundley 11 theorem 2.2.12)

This adelic integral-method generalizes to Dedekind zeta functions for any algebraic number field. This is due to (Tate 50), highlighted in (Goldfeld-Hundley 11, Remark (1) by Ivan Fesenko).

Euler product form

…Euler product…

$$\sum_{n=1}^\infty \frac{1}{n^s} = \underset{p\; prime}{\prod} \frac{1}{1-p^{-s}}$$

Analogs over number fields, function fields and complex curves

The function field analogy in view of the discussion at zeta function of an elliptic differential operator says that the Riemann zeta function is analogous to the regulated functional trace of a would-be “Dirac operator on Spec(Z)”.

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

Other identifications/analogies of the Riemann zeta function (and more generally the Dedekind zeta-function) with partition functions in physics have been proposed, in particular the Bost-Connes system.

See also at function field analogy.

Related concepts

• Riemann hypothesis

References

Discussion in the context of adelic integration and higher arithmetic geometry is in

• John Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950

• Ivan Fesenko, Adelic approch to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273–317 (pdf)

with review including

• Paul Garrett, Iwasawa-Tate on ζ-functions and L-functions, 2011 (pdf

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

• 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)

Discussion in the context of p-adic string theory:

• An Huang, Bogdan Stoica, Shing-Tung Yau, General relativity from $p$-adic strings (arXiv:1901.02013)