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

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 ζ ^ K (s) 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=, where for w=1 this function is ζ ^(s)=π -s 2 Γ(s 2)ζ(s). The function ξ (w,s):=s(s-w) 2wZ (w,s) is shown to be an entire function on 2 , to satisfy the functional equation ξ (w,s)=ξ (w,w-s), and to have ξ (0,s)=-s 2 8(1-2 1+s 2 )(1-2 1-s 2 )ζ ^(s 2)ζ ^(-s 2). We study the location of the zeros of Z (w,s) for various real values of w=u. For fixed u0 the zeros are confined to a vertical strip of width at most u+16 and the number of zeros N u (T) to height T has similar asymptotics to the Riemann zeta function. For fixed u<0 these functions are strictly positive on the "critical line" (s)=u 2. This phenomenon is associated to a positive convolution semigroup with parameter u >0 , which is a semigroup of infinitely divisible probability distributions, having densities P u (x)dx for real x, where P u (x)=1 2πθ(1) u Z (-u,-u 2+ix), and θ(1)=π 1/4 /Γ(3/4).

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 ζ ^ K (s) 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=, pour lequel lorsque w=1 la fonction est ζ ^(s)=π -s 2 Γ(s 2)ζ(s). Nous montrons que la fonction ξ (w,s):=s(s-w) 2wZ (w,s) est une fonction entière sur 2 , satisfaisant l’équation fonctionnelle ξ (w,s)=ξ (w,w-s), et vérifiant ξ (0,s)=-s 2 8(1-2 1+s 2 )(1-2 1-s 2 )ζ ^(s 2)ζ ^(-s 2). Nous étudions l’emplacement des zéros de Z (w,s) pour les valeurs réelles de w=u. Pour u0 fixé, les zéros sont situés dans une bande verticale de largeur au plus u+16 et le nombre N u (T) 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 (u,s) sont strictement positives sur la "droite critique" (s)=u 2. Ce phénomène est associé à un semi-groupe de convolution, positif, de paramètre u >0 , qui est un semi-groupe de lois de probabilités infiniment divisibles, ayant les densités P u (x)dx pour x réel, avec P u (x)=1 2πθ(1) u Z (-u,-u 2+ix), et θ(1)=π 1/4 /Γ(3/4).

DOI: 10.5802/aif.1939
Classification: 11M41,  11G40,  60E07
Keywords: Arakelov divisors, functional equation, infinitely divisible distributions, zeta functions
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     title = {On a two-variable zeta function for number fields},
     journal = {Annales de l'Institut Fourier},
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     volume = {53},
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     year = {2003},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1939/

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