Cet article étudie une fonction zêta à deux variables attachée à un corps de nombres algébriques . 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 cette fonction devient la fonction zêta de Dedekind complète du corps . C’est une fonction méromorphe de deux variables complexes avec comme diviseur des pôles, et elle satisfait l’équation fonctionnelle . Nous considérons le cas particulier , pour lequel lorsque la fonction est . Nous montrons que la fonction est une fonction entière sur , satisfaisant l’équation fonctionnelle et vérifiant Nous étudions l’emplacement des zéros de pour les valeurs réelles de . Pour fixé, les zéros sont situés dans une bande verticale de largeur au plus et le nombre de zéros de hauteurs au plus possède une asymptotique semblable à celle s’appliquant aux zéros de la fonction zêta de Riemann. Pour , les fonctions sont strictement positives sur la “droite critique” . Ce phénomène est associé à un semi-groupe de convolution, positif, de paramètre , qui est un semi-groupe de lois de probabilités infiniment divisibles, ayant les densités pour réel, avec et
This paper studies a two-variable zeta function attached to an algebraic number field , 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 this function becomes the completed Dedekind zeta function of the field . The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the special case , where for this function is . The function is shown to be an entire function on , to satisfy the functional equation and to have We study the location of the zeros of for various real values of . For fixed the zeros are confined to a vertical strip of width at most and the number of zeros to height has similar asymptotics to the Riemann zeta function. For fixed these functions are strictly positive on the “critical line” . This phenomenon is associated to a positive convolution semigroup with parameter , which is a semigroup of infinitely divisible probability distributions, having densities for real , where and .
Keywords: Arakelov divisors, functional equation, infinitely divisible distributions, zeta functions
Mot clés : diviseurs d'Arakelov, équation fonctionnelle, lois de probabilités infiniment divisibles, fonction zêta
@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}, pages = {1--68}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {1}, year = {2003}, doi = {10.5802/aif.1939}, mrnumber = {1973068}, zbl = {1106.11036}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1939/} }
TY - JOUR AU - Lagarias, Jeffrey C. AU - Rains, Eric TI - On a two-variable zeta function for number fields JO - Annales de l'Institut Fourier PY - 2003 SP - 1 EP - 68 VL - 53 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1939/ DO - 10.5802/aif.1939 LA - en ID - AIF_2003__53_1_1_0 ER -
%0 Journal Article %A Lagarias, Jeffrey C. %A Rains, Eric %T On a two-variable zeta function for number fields %J Annales de l'Institut Fourier %D 2003 %P 1-68 %V 53 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1939/ %R 10.5802/aif.1939 %G en %F 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, Tome 53 (2003) no. 1, pp. 1-68. doi : 10.5802/aif.1939. http://www.numdam.org/articles/10.5802/aif.1939/
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