On a two-variable zeta function for number fields

Lagarias, Jeffrey C., and Rains, Eric. "On a two-variable zeta function for number fields." Annales de l’institut Fourier 53.1 (2003): 1-68. <http://eudml.org/doc/116034>.

@article{Lagarias2003,
abstract = {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 $\hat\{\zeta \}_K(s)$ 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 $K = \{\mathbb \{Q\}\}$, where for $w = 1$ this function is $\hat\{\zeta \}(s)= \pi ^\{- \{s\over 2\}\} \Gamma (\{s\over 2\}) \zeta (s)$. The function $\xi _\{\{\mathbb \{Q\}\}\}(w, s) := \{s(s- w)\over 2 w\}Z_\{\{\mathbb \{Q\}\}\}(w, s)$ is shown to be an entire function on $\{\mathbb \{C\}\}^2$, to satisfy the functional equation $\xi _\{\{\mathbb \{Q\}\}\}(w, s) = \xi _\{\{\mathbb \{Q\}\}\}(w, w - s),$ and to have $\xi _\{\{\mathbb \{Q\}\}\}(0, s) =-\{s^2\over 8\}(1 - 2^\{1 + \{s\over 2\}\}) (1 - 2^\{1 - \{s\over 2\}\}) \hat\{\zeta \}(\{s\over 2\}) \hat\{\zeta \}(\{-s\over 2\}).$ 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 $N_u(T)$ to height $T$ has similar asymptotics to the Riemann zeta function. For fixed $u &lt; 0$ these functions are strictly positive on the “critical line” $\{\mathfrak \{R\}\}(s) = \{u\over 2\}$. This phenomenon is associated to a positive convolution semigroup with parameter $u \in \{\mathbb \{R\}\}_\{&gt; 0\}$, which is a semigroup of infinitely divisible probability distributions, having densities $P_u(x)dx$ for real $x$, where $P_u(x) = \{1\over 2\pi \}\theta (1)^u Z_\{\{\mathbb \{Q\}\}\}(-u, -\{u\over 2\} + ix),$ and $\theta (1) = \pi ^\{1/4\}/\Gamma (3/4)$.},
affiliation = {AT & T Labs - Research, Florham Park NJ 07932 (USA); Center for Communications Research, 805 Bunn Drive, Princeton NJ 09540 (USA)},
author = {Lagarias, Jeffrey C., Rains, Eric},
journal = {Annales de l’institut Fourier},
keywords = {Arakelov divisors; functional equation; infinitely divisible distributions; zeta functions},
language = {eng},
number = {1},
pages = {1-68},
publisher = {Association des Annales de l'Institut Fourier},
title = {On a two-variable zeta function for number fields},
url = {http://eudml.org/doc/116034},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 1
SP - 1
EP - 68
AB - 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 $\hat{\zeta }_K(s)$ 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 $K = {\mathbb {Q}}$, where for $w = 1$ this function is $\hat{\zeta }(s)= \pi ^{- {s\over 2}} \Gamma ({s\over 2}) \zeta (s)$. The function $\xi _{{\mathbb {Q}}}(w, s) := {s(s- w)\over 2 w}Z_{{\mathbb {Q}}}(w, s)$ is shown to be an entire function on ${\mathbb {C}}^2$, to satisfy the functional equation $\xi _{{\mathbb {Q}}}(w, s) = \xi _{{\mathbb {Q}}}(w, w - s),$ and to have $\xi _{{\mathbb {Q}}}(0, s) =-{s^2\over 8}(1 - 2^{1 + {s\over 2}}) (1 - 2^{1 - {s\over 2}}) \hat{\zeta }({s\over 2}) \hat{\zeta }({-s\over 2}).$ 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 $N_u(T)$ to height $T$ has similar asymptotics to the Riemann zeta function. For fixed $u &lt; 0$ these functions are strictly positive on the “critical line” ${\mathfrak {R}}(s) = {u\over 2}$. This phenomenon is associated to a positive convolution semigroup with parameter $u \in {\mathbb {R}}_{&gt; 0}$, which is a semigroup of infinitely divisible probability distributions, having densities $P_u(x)dx$ for real $x$, where $P_u(x) = {1\over 2\pi }\theta (1)^u Z_{{\mathbb {Q}}}(-u, -{u\over 2} + ix),$ and $\theta (1) = \pi ^{1/4}/\Gamma (3/4)$.
LA - eng
KW - Arakelov divisors; functional equation; infinitely divisible distributions; zeta functions
UR - http://eudml.org/doc/116034
ER -