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 .
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
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, Volume 53 (2003) no. 1, pp. 1-68. doi : 10.5802/aif.1939. http://www.numdam.org/articles/10.5802/aif.1939/
[1] The Theory of Partitions, Addison-Wesley (Reprint: Cambridge University Press, 1998), Reading, Mass., 1976 | MR | Zbl
[2] Special Functions, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl
[3] Modular Functions and Dirichlet Series in Number Theory, Springer, New York, 1976 | MR | Zbl
[4] Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc, Volume 38 (2001), pp. 435-465 | DOI | MR | Zbl
[5] On the distribution of zeros of linear combinations of Euler products, Duke Math. J, Volume 80 (1995), pp. 821-862 | MR | Zbl
[6] Convolution structures and arithmetic cohomology (3 Jan 2001) (e-print, arXiv: math.AG9807151 v3)
[7] Families of Automorphic Forms, Birkhäuser Verlag, Basel, 1994 | MR | Zbl
[8] Turán inequalitites and zeros of Dirichlet series associated with certain cusp forms, Trans. Amer. Math. Soc, Volume 342 (1994), pp. 407-419 | DOI | MR | Zbl
[9] Multiplicative Number Theory, Springer-Verlag, New York, 1980 | MR | Zbl
[10] An Introduction to Probability Theory and its Applications, Volume II, John Wiley \& Sons, New York, 1971 | MR | Zbl
[11] Effectivity of Arakelov Divisors and the theta divisor of a number field, Selecta Math., New Series, 6, eprint: \tt arXiv math.AG/9802121, 2000 | MR | Zbl
[12] On a result of Selberg concerning zeros of linear combinations of L-functions, Internat. Mat. Research Notices (2000) no. 11, pp. 551-577 | DOI | MR | Zbl
[13] Introduction to Modular Forms, Springer-Verlag, New York, 1976 | MR | Zbl
[14] Algebraic Number Theory, Springer-Verlag, New York, 1994 | MR | Zbl
[15] The Fourier coefficients of automorphic forms on horocyclic groups II, Michigan Math. J, Volume 6 (1959), pp. 173-193 | DOI | MR | Zbl
[16] Magnitude of the Fourier coefficients of automorphic forms of negative dimension, Bull. Amer. Math. Soc, Volume 67 (1961), pp. 603-606 | DOI | MR | Zbl
[17] Discontinuous Groups and Arithmetic Subgroups, Mathematical Surveys, Number VIII, Amer. Math. Soc., Providence, RI, 1964 | Zbl
[18] On special divisors and the two variable zeta function of algebraic curves over finite fields, Arithmetic, Geometry and Coding Theory (1996), pp. 175-184 | Zbl
[19] Über automorphe Orthogonalfunktionen und die Konstruktion der automorphen Formen von positiver reeller Dimension, Math. Ann, Volume 127 (1954), pp. 33-81 | DOI | MR | Zbl
[20] Über Betragmittelwerte und die Fourier-Koeffizienten der ganzen automorphen Formen, Arch. Math. (Basel), Volume 9 (1958), pp. 176-182 | MR | Zbl
[21] Infinitely divisible laws associated to hyperbolic functions, Univ. Calif.-Berkeley Stat. Technical Rept. (2001) no. 581
[22] On the expansion of the partition function in a series, Ann. Math, Volume 44 (1943), pp. 416-422 | DOI | MR | Zbl
[23] Introduction to the Theory of Entire Functions of Several Variables, Amer. Math. Soc., Providence, RI, 1974 | MR | Zbl
[24] Entire Functions, Several Complex Variables III (Encyclopedia of Mathematical Sciences), Volume Volume 9 (1989), pp. 1-30
[25] Holomorphic Functions of Finite Order in Several Complex Variables, CBMS Publication, Volume No. 21 (1974) | Zbl
Cited by Sources: