# ANNALES DE L'INSTITUT FOURIER

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 ${Z}_{K}\left(w,s\right)$ 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: 10.5802/aif.1939
Classification: 11M41, 11G40, 60E07
Keywords: Arakelov divisors, functional equation, infinitely divisible distributions, zeta functions
Mot clés : diviseurs d'Arakelov, équation fonctionnelle, lois de probabilités infiniment divisibles, fonction zêta
Lagarias, Jeffrey C. 1; Rains, Eric 2

1 AT \& T Labs - Research, Florham Park NJ 07932 (USA)
2 Center for Communications Research, 805 Bunn Drive, Princeton NJ 09540 (USA)
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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/

[1] G. E. Andrews The Theory of Partitions, Addison-Wesley (Reprint: Cambridge University Press, 1998), Reading, Mass., 1976 | MR | Zbl

[2] G. E. Andrews; R. Askey; R. Roy Special Functions, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[3] T. M. Apostol Modular Functions and Dirichlet Series in Number Theory, Springer, New York, 1976 | MR | Zbl

[4] P. Biane; J. Pitman; M. Yor 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] E. Bombieri; D. A. Hejhal On the distribution of zeros of linear combinations of Euler products, Duke Math. J, Volume 80 (1995), pp. 821-862 | MR | Zbl

[6] A. Borisov Convolution structures and arithmetic cohomology (3 Jan 2001) (e-print, arXiv: math.AG9807151 v3)

[7] R. W. Bruggeman Families of Automorphic Forms, Birkhäuser Verlag, Basel, 1994 | MR | Zbl

[8] J. B. Conrey; A. Ghosh 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] H. Davenport Multiplicative Number Theory, Springer-Verlag, New York, 1980 | MR | Zbl

[10] W. Feller An Introduction to Probability Theory and its Applications, Volume II, John Wiley \& Sons, New York, 1971 | MR | Zbl

[11] G. van der Geer; R. Schoof 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] D. A. Hejhal 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] S. Lang Introduction to Modular Forms, Springer-Verlag, New York, 1976 | MR | Zbl

[14] S. Lang Algebraic Number Theory, Springer-Verlag, New York, 1994 | MR | Zbl

[15] J. Lehner The Fourier coefficients of automorphic forms on horocyclic groups II, Michigan Math. J, Volume 6 (1959), pp. 173-193 | DOI | MR | Zbl

[16] J. Lehner 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] J. Lehner Discontinuous Groups and Arithmetic Subgroups, Mathematical Surveys, Number VIII, Amer. Math. Soc., Providence, RI, 1964 | Zbl

[18] R. Pellikaan; R. Pellikaan, M. Perret and S. G. Vladut 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] H. Petersson 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] H. Petersson Über Betragmittelwerte und die Fourier-Koeffizienten der ganzen automorphen Formen, Arch. Math. (Basel), Volume 9 (1958), pp. 176-182 | MR | Zbl

[21] J. Pitman; M. Yor Infinitely divisible laws associated to hyperbolic functions, Univ. Calif.-Berkeley Stat. Technical Rept. (2001) no. 581

[22] H. Rademacher On the expansion of the partition function in a series, Ann. Math, Volume 44 (1943), pp. 416-422 | DOI | MR | Zbl

[23] L. I. Ronkin Introduction to the Theory of Entire Functions of Several Variables, Amer. Math. Soc., Providence, RI, 1974 | MR | Zbl

[24] L. I. Ronkin; G. M. Khenkin, Ed. Entire Functions, Several Complex Variables III (Encyclopedia of Mathematical Sciences), Volume Volume 9 (1989), pp. 1-30

[25] W. Stoll Holomorphic Functions of Finite Order in Several Complex Variables, CBMS Publication, Volume No. 21 (1974) | Zbl

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