Zeta functions, Mellin transforms and the Gelfand-Leray form

We outline the similarities between zeta functions appearing in number theory and in statistical learning.

This post is a continuation from our series on spiking networks, path integrals and motivic information.

Gelfand-Leray functions

Let \(\omega\) be a volume form or a form resulting from a measure on some space \(W\). Let \(f: W \rightarrow \mathbb{R}\) be a non-negative energy function or a Kullback-Leibler divergence. A Gelfand-Leray form of \(\omega\) is any differential form \(\phi\) that satisfies

\[df \wedge \phi = \omega.\]

We will denote \(\phi\) by \(\omega /df\). See [AGZV88] and §4 of [Mar10] for expositions about the Gelfand-Leray form.

The Gelfand-Leray function is

\[J(t) =\displaystyle \int_{f(w)=t} \frac{\omega}{df}.\]

These functions often have asymptotic expansions of the form

\[J(t) = \displaystyle \sum_\alpha \sum_{i=1}^d m_{\alpha,i} t^\alpha (\log t)^{i-1}.\]

Here, the \(\alpha\)’s are poles and the \(i\)’s are their multiplicities.

Zeta functions

Informally, zeta functions are complex functions of the form

\[\zeta(z) = \displaystyle \sum_\lambda m_\lambda \lambda^{-z}\]

where the sum is over some possibly infinite (multi-)set of complex numbers \(\lambda\) and the coefficients \(m_\lambda\) are complex.

For example, the Riemann zeta function is

\[\zeta(z) = \displaystyle \sum_{n=1}^\infty n^{-z}.\]

Zeta functions often have analytic continuations to the whole complex plane, and possess Laurent expansions of the form

\[\zeta(z) = \zeta_0(z) + \sum_\alpha \sum_{i=1}^d c_{\alpha i} (z-\alpha)^{-i}\]

where \(\zeta_0(z)\) is holomorphic and the \(\alpha\) are the poles.

A zeta function can often be written as the Mellin transform

\[\zeta(z) = \displaystyle \int_0^\infty t^{z-1} J(t) dt = \int_W f(w)^{z-1} \omega\]

of a function \(J(t)\), and \(J(t)\) is often some kind of Gelfand-Leray function.

For example, \(\zeta(z) = \sum_\lambda m_\lambda \lambda^{-z}\) is the Mellin transform

\[\zeta(z) = \displaystyle \frac{1}{\Gamma(s)} \int_0^\infty \theta(t) t^{z-1} dt\]

of the theta function

\[\theta(t) = \displaystyle \sum_{\lambda} m_\lambda e^{- \lambda t}.\]

Here, if we expand each \(e^{-\lambda t}\) by its Taylor series, we see that we can think of it as a kind of Gelfand-Leray function with non-negative integer poles and unit multiplicities.

However, I do not know if the theta function can actually be expressed as the Gelfand-Leray function of a differential form with respect to an energy function.

Zeta functions are also useful for defining infinite products [Man95]

\[Z = \displaystyle \prod_\lambda \lambda^{m_\lambda}\]

which we define to be

\[Z = \exp \left( - \zeta'(0)\right)\]

where

\[\zeta(z) = \displaystyle \sum_\lambda m_\lambda \lambda^{-z}.\]

Indeed, for finite sums, we see that

\[\zeta'(0) = - \displaystyle \sum_\lambda m_\lambda \log \lambda.\]

Statistical Learning

In statistical learning, given a prior \(\omega = \psi(w)dw\) on the parameters \(w\) of a statistical model \(p: W \rightarrow \Delta,\) and given the relative information \(K(w)\) to the true distribution \(q\) from the model distribution \(p(w),\) the Gelfand-Leray function

\[J(t) = \displaystyle \frac{d}{dt} \int_{0 \leq K(w) \leq t} \psi(w) dw\]

is called the state density function [Wat09].

The zeta function of the statistical model is

\[\zeta(z) = \int_W K(w)^z \psi(w) dw = \int_0^\infty t^z J(t)dt\]

which (up to some linear change of variable in \(z\)) is the Mellin transform of the state density function \(J(t).\)

Number Theory

In number theory, we have the notorious Riemann zeta function which becomes more interesting [Bae05] if we write it as

\[\zeta(z) = \displaystyle \sum_{n=1}^\infty \left(n^2\right)^{-\frac{z}{2}}.\]

This representation suggests that we study the infamous Jacobi theta function

\[\theta(t) =\displaystyle \sum_{n=-\infty}^\infty e^{-n^2 (\pi t) }.\]

Note the extra factor of \(\pi\) in the exponent and the changing of the lower bound of summation to \(-\infty\).

The modularity of the theta function

\[\theta(t) = \displaystyle \frac{1}{\sqrt{t}} \theta\left(\frac{1}{t}\right)\]

is the crux behind the functional equation for the Riemann zeta function

\[\Lambda(z) = \Lambda(1-z)\]

where \(\Lambda(z) = 2^{-1/2}\pi^{-z/2}\Gamma(z/2)\zeta(z).\)

The function \(\Lambda(z)\) itself can be written [Den91] as the infinite product

\[(\frac{z}{2\pi})(\frac{z-1}{2\pi})\Lambda(z) = \prod_\rho \frac{z-\rho}{2\pi}\]

over the zeros of the Riemann zeta function. This infinite product is in turn expressible using another zeta function and theta function.

The Hurwitz zeta function

\[\zeta_s(z) = \displaystyle \sum_{n=0}^\infty (n+s)^{-z}\]

is associated to the theta function \(\theta_s(t) = e^{-st} \theta(t)\) where

\[\theta(t) =\sum_{n=0}^\infty e^{-nt} = \frac{1}{1-e^{-t}}.\]

Note that the Riemann zeta function is the special case \(\zeta_1(z).\)

The Hasse-Weil zeta function of a smooth absolutely irreducible curve \(V\) over a finite field \(\mathbb{F}_q\) is

\[\begin{split}\begin{array}{rl} Z_{V}(z) &= \displaystyle \sum_\alpha \left(q^{ \deg(\alpha)}\right)^{-z} \\ & \\ &= \exp\left(\displaystyle\sum_{m\geq 1} \frac{\# V(\mathbb{F}_{q^m})}{m}q^{-zm} \right) \end{array}\end{split}\]

where the first sum is over effective zero-cycles \(\alpha\). The zeta function can also be written as

\[(1-q^{-z})(1-q^{1-z})Z_{V}(z) = \displaystyle \prod_{j=1}^{2g} (1-\phi_j q^{-z})\]

where \(g\) is the genus of \(V\) and each \(\phi_j\) is an algebraic integer. Each of the factors can be written as infinite products

\[1-\lambda q^{-z} = \displaystyle \prod_{\rho \vert q^\rho = \lambda} \frac{\log q}{2\pi i}(z-\rho).\]

which is in turn expressible using another zeta function and theta function.

References

[AGZV88]

VI Arnold, SM Gusein-Zade, and AN Varchenko. Elementary integrals and the resolution of singularities of the phase. In Singularities of Differentiable Maps, pages 215–232. Springer, 1988.

[Bae05]

John Baez. This week’s finds in mathematical physics (week 217). https://math.ucr.edu/home/baez/week217.html, 2005.

[Den91]

Christopher Deninger. On the γ-factors attached to motives. Inventiones mathematicae, 104(1):245–261, 1991.

[Man95]

Yuri Manin. Lectures on zeta functions and motives. Astérisque, 228:121–163, 1995.

[Mar10]

Matilde Marcolli. Feynman motives. World Scientific, 2010.

[Wat09]

Sumio Watanabe. Algebraic geometry and statistical learning theory. Number 25. Cambridge university press, 2009.

Conditional relative information and its axiomatizations   Motivic relative information