On a two-variable zeta function for number fields

Lagarias, Jeffrey C.; Rains, Eric

Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 1-68.

Cet article étudie une fonction zêta à deux variables $Z_K(w,s)$ 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 ${\widehat{\zeta }}_K\left(s\right)$ du corps $K$. C’est une fonction méromorphe de deux variables complexes avec $s(w-s)$ comme diviseur des pôles, et elle satisfait l’équation fonctionnelle $Z_K(w,s)=Z_K(w,w-s)$. Nous considérons le cas particulier $K=\mathbb{Q}$, pour lequel lorsque $w=1$ la fonction est $\widehat{\zeta }\left(s\right)={\pi }^{-\frac{s}{2}}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right)$. Nous montrons que la fonction ${\xi }_{\mathbb{Q}}(w,s):=\frac{s(s-w)}{2w}{Z}_{\mathbb{Q}}(w,s)$ est une fonction entière sur ${ℂ}^2$, satisfaisant l’équation fonctionnelle ${\xi }_{\mathbb{Q}}(w,s)={\xi }_{\mathbb{Q}}(w,w-s),$ et vérifiant ${\xi }_{\mathbb{Q}}(0,s)=-\frac{s^2}{8}(1-2^{1+\frac{s}{2}})(1-2^{1-\frac{s}{2}})\widehat{\zeta }\left(\frac{s}{2}\right)\widehat{\zeta }\left(\frac{-s}{2}\right).$ Nous étudions l’emplacement des zéros de $Z_{\mathbb{Q}}(w,s)$ 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 $Z_{\mathbb{Q}}(u,s)$ 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 $P_u\left(x\right)=\frac{1}{2\pi }\theta {\left(1\right)}^u{Z}_{\mathbb{Q}}(-u,-\frac{u}{2}+ix),$ et $\theta \left(1\right)={\pi }^{1/4}/\Gamma (3/4).$

This paper studies a two-variable zeta function $Z_K(w,s)$ 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. When $w=1$ this function becomes the completed Dedekind zeta function ${\widehat{\zeta }}_K\left(s\right)$ of the field $K$. The function is a meromorphic function of two complex variables with polar divisor $s(w-s)$, and it satisfies the functional equation $Z_K(w,s)=Z_K(w,w-s)$. We consider the special case $K=\mathbb{Q}$, where for $w=1$ this function is $\widehat{\zeta }\left(s\right)={\pi }^{-\frac{s}{2}}\Gamma \left(\frac{s}{2}\right)\zeta \left(s\right)$. The function ${\xi }_{\mathbb{Q}}(w,s):=\frac{s(s-w)}{2w}{Z}_{\mathbb{Q}}(w,s)$ is shown to be an entire function on ${ℂ}^2$, to satisfy the functional equation ${\xi }_{\mathbb{Q}}(w,s)={\xi }_{\mathbb{Q}}(w,w-s),$ and to have ${\xi }_{\mathbb{Q}}(0,s)=-\frac{s^2}{8}(1-2^{1+\frac{s}{2}})(1-2^{1-\frac{s}{2}})\widehat{\zeta }\left(\frac{s}{2}\right)\widehat{\zeta }\left(\frac{-s}{2}\right).$ We study the location of the zeros of $Z_{\mathbb{Q}}(w,s)$ 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 $P_u\left(x\right)=\frac{1}{2\pi }\theta {\left(1\right)}^u{Z}_{\mathbb{Q}}(-u,-\frac{u}{2}+ix),$ and $\theta \left(1\right)={\pi }^{1/4}/\Gamma (3/4)$.

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