On a two-variable zeta function for number fields
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, p. 1-68

This paper studies a two-variable zeta function ${Z}_{K}\left(w,s\right)$ 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. When $w=1$ this function becomes the completed Dedekind zeta function ${\stackrel{^}{\zeta }}_{K}\left(s\right)$ of the field $K$. The function is a meromorphic function of two complex variables with polar divisor $s\left(w-s\right)$, and it satisfies the functional equation ${Z}_{K}\left(w,s\right)={Z}_{K}\left(w,w-s\right)$. We consider the special case $K=ℚ$, where for $w=1$ this function is $\stackrel{^}{\zeta }\left(s\right)={\pi }^{-\frac{s}{2}}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right)$. The function ${\xi }_{ℚ}\left(w,s\right):=\frac{s\left(s-w\right)}{2w}{Z}_{ℚ}\left(w,s\right)$ is shown to be an entire function on ${ℂ}^{2}$, to satisfy the functional equation ${\xi }_{ℚ}\left(w,s\right)={\xi }_{ℚ}\left(w,w-s\right),$ and to have ${\xi }_{ℚ}\left(0,s\right)=-\frac{{s}^{2}}{8}\left(1-{2}^{1+\frac{s}{2}}\right)\left(1-{2}^{1-\frac{s}{2}}\right)\stackrel{^}{\zeta }\left(\frac{s}{2}\right)\stackrel{^}{\zeta }\left(\frac{-s}{2}\right).$ We study the location of the zeros of ${Z}_{ℚ}\left(w,s\right)$ for various real values of $w=u$. For fixed $u\ge 0$ the zeros are confined to a vertical strip of width at most $u+16$ and the number of zeros ${N}_{u}\left(T\right)$ to height $T$ has similar asymptotics to the Riemann zeta function. For fixed $u<0$ these functions are strictly positive on the “critical line” $\Re \left(s\right)=\frac{u}{2}$. This phenomenon is associated to a positive convolution semigroup with parameter $u\in {ℝ}_{>0}$, which is a semigroup of infinitely divisible probability distributions, having densities ${P}_{u}\left(x\right)dx$ for real $x$, where ${P}_{u}\left(x\right)=\frac{1}{2\pi }\theta {\left(1\right)}^{u}{Z}_{ℚ}\left(-u,-\frac{u}{2}+ix\right),$ and $\theta \left(1\right)={\pi }^{1/4}/\Gamma \left(3/4\right)$.

Cet article étudie une fonction zêta à deux variables ${Z}_{K}\left(w,s\right)$ attachée à un corps de nombres algébriques $K$. Définie par van der Geer et Schoof, elle provient d’un analogue du théorème de Riemann-Roch pour les corps de nombres, utilisant les diviseurs d’Arakelov. Lorsque $w=1$ cette fonction devient la fonction zêta de Dedekind complète ${\stackrel{^}{\zeta }}_{K}\left(s\right)$ du corps $K$. C’est une fonction méromorphe de deux variables complexes avec $s\left(w-s\right)$ comme diviseur des pôles, et elle satisfait l’équation fonctionnelle ${Z}_{K}\left(w,s\right)={Z}_{K}\left(w,w-s\right)$. Nous considérons le cas particulier $K=ℚ$, pour lequel lorsque $w=1$ la fonction est $\stackrel{^}{\zeta }\left(s\right)={\pi }^{-\frac{s}{2}}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right)$. Nous montrons que la fonction ${\xi }_{ℚ}\left(w,s\right):=\frac{s\left(s-w\right)}{2w}{Z}_{ℚ}\left(w,s\right)$ est une fonction entière sur ${ℂ}^{2}$, satisfaisant l’équation fonctionnelle ${\xi }_{ℚ}\left(w,s\right)={\xi }_{ℚ}\left(w,w-s\right),$ et vérifiant ${\xi }_{ℚ}\left(0,s\right)=-\frac{{s}^{2}}{8}\left(1-{2}^{1+\frac{s}{2}}\right)\left(1-{2}^{1-\frac{s}{2}}\right)\stackrel{^}{\zeta }\left(\frac{s}{2}\right)\stackrel{^}{\zeta }\left(\frac{-s}{2}\right).$ Nous étudions l’emplacement des zéros de ${Z}_{ℚ}\left(w,s\right)$ pour les valeurs réelles de $w=u$. Pour $u\ge 0$ fixé, les zéros sont situés dans une bande verticale de largeur au plus $u+16$ et le nombre ${N}_{u}\left(T\right)$ de zéros de hauteurs au plus $T$ possède une asymptotique semblable à celle s’appliquant aux zéros de la fonction zêta de Riemann. Pour $u<0$, les fonctions ${Z}_{ℚ}\left(u,s\right)$ sont strictement positives sur la “droite critique” $\Re \left(s\right)=\frac{u}{2}$. Ce phénomène est associé à un semi-groupe de convolution, positif, de paramètre $u\in {ℝ}_{>0}$, qui est un semi-groupe de lois de probabilités infiniment divisibles, ayant les densités ${P}_{u}\left(x\right)dx$ pour $x$ réel, avec ${P}_{u}\left(x\right)=\frac{1}{2\pi }\theta {\left(1\right)}^{u}{Z}_{ℚ}\left(-u,-\frac{u}{2}+ix\right),$ et $\theta \left(1\right)={\pi }^{1/4}/\Gamma \left(3/4\right).$

DOI : https://doi.org/10.5802/aif.1939
Classification:  11M41,  11G40,  60E07
Keywords: Arakelov divisors, functional equation, infinitely divisible distributions, zeta functions
@article{AIF_2003__53_1_1_0,
author = {Lagarias, Jeffrey C. and Rains, Eric},
title = {On a two-variable zeta function for number fields},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {53},
number = {1},
year = {2003},
pages = {1-68},
doi = {10.5802/aif.1939},
zbl = {1106.11036},
mrnumber = {1973068},
language = {en},
url = {http://www.numdam.org/item/AIF_2003__53_1_1_0}
}

Lagarias, Jeffrey C.; Rains, Eric. On a two-variable zeta function for number fields. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 1-68. doi : 10.5802/aif.1939. http://www.numdam.org/item/AIF_2003__53_1_1_0/

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