On some families of divisible formal weight enumerators and their zeta functions

Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility

Let $a(1) >0$, $a(n) \ge 0$ for $n \ge 2$ and $a(n) = O(n^\varepsilon)$ for any $\varepsilon >0$, and put $Z(σ+ it):= \sum_{n=1}^\infty a(n) n^{-σ- it}$ where $σ, t \in {\mathbb{R}}$. In the present paper, we show that any zeta distribution whose characteristic function is defined by ${\mathcal{Z}}_σ(t) :=Z(σ+ it)/Z(σ)$ is pretended infinitely divisible if $σ>1$ is sufficiently large. Moreover, we prove that if ${\mathcal{Z}}_σ(t)$ is an infinitely divisible characteristic function for some $σ_{id} >1$, then ${\mathcal{Z}}_σ(t)$ is infinitely divisible for all $σ>1$. Note that the correspondin…

Results on zeta functions for codes

We give a new and short proof of the Mallows-Sloane upper bound for self-dual codes. We formulate a version of Greene's theorem for normalized weight enumerators. We relate normalized rank-generating polynomials to two-variable zeta functions. And we show that a self-dual code has the Clifford property, but that the same property does not hold in general for formally self-dual codes.

关于与黎曼假设相关的无限可分分布

On infinitely divisible distributions related to the Riemann hypothesis

本研究引入一个与黎曼ζ函数相关的函数，建立了该函数作为实轴上无限可分分布特征函数的充要条件与黎曼假设成立的等价关系。

Multidimensional Shintani zeta functions and zeta distributions on R^d

The class of Riemann zeta distribution is one of the classical classes of probability distributions on R. Multidimensional Shintani zeta function is introduced and its definable probability distributions on R^d are studied. This class contains some fundamental probability distributions such as binomial and Poisson distributions. The relation with multidimensional polynomial Euler product, which induces multidimensional infinitely divisible distributions on R^d, is also studied.

An abundance of invariant polynomials satisfying the Riemann hypothesis

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It can also be defined for other homogeneous polynomials not corresponding to existing codes. If the homogeneous polynomial is invariant under the MacWilliams transform, then its zeta function satisfies a functional equation and we can formulate an analogue of the Riemann hypothesis. As far as existing codes are concerned, the Riemann hypothesis is believed to be closely related to the extremal property. In this article, we show there are abundant polynomials invariant b…

Behaviors of multivariable finite Euler products in probabilistic view

Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include functions which generate infinitely divisible, not infinitely divisible characteristic functions and not even to be characteristic functions. In this paper, we treat some multivariable finite Euler products and show how they behave in view of such properties.

Complete determination of the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus

We compute the coefficients of the polynomials $C_n(q)$ defined by the equation \begin{equation*} 1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}} \, . \end{equation*} As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials $C_n(q)$ are divisible by $(q-1)^2$. We also compute the coefficients of the polynomials $P_n(q) = C_n(q)/(q-1)^2$: each coefficient counts the …

Lower bound for the poles of Igusa's p-adic zeta functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n>1 variables and let χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we …

有限格的概率 Zeta 函数

The Probabilistic Zeta Function of a Finite Lattice

本文研究有限格的概率 zeta 函数，给出一般 Dirichlet 级数表示，计算典型格并联系第二类 Stirling 数，还利用素数定理证明 $d$-可整除划分格的该函数通常不是通常 Dirichlet 级数。

On a two-variable zeta function for number fields

This paper studies a zeta function of two complex variables (w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. We mainly consider the case of the rational field Q, where for w = 1 one recovers the Riemann zeta function with the factors at infinity added. The analogue of the Riemann xi-function analytically continues to an entire function of two complex variables, and satisfies a functional equation holding w fixed and sending s to w-s. For real w the "critica…

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $σ$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $ζ_σ$ of $σ$ is either rational or transcendental, the first case happening precisely when $σ^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $σ$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_σ$, in which the number of fixed points of $σ^n$ is replaced by the degree of $σ^n-1$, is always a rational funct…