scholarly journals Zeta functions of the 3-dimensional almost-Bieberbach groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego Sulca
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Euler Product ◽  
3 Dimensional ◽  
Local Factors ◽  
Local Functional ◽  
Finite Sums ◽  
Complete Computation ◽  
Local Functional Equations ◽  
Torsion Free

Abstract The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

scholarly journals A nilpotent group without local functional equations for pro-isomorphic subgroups

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Mark N. Berman ◽  
Benjamin Klopsch
Keyword(s):  
Zeta Function ◽  
Nilpotent Group ◽  
Functional Equations ◽  
Finitely Generated ◽  
Local Functional ◽  
Local Functional Equations ◽  
Torsion Free

AbstractThe pro-isomorphic zeta function ζWe manufacture the first example of a torsion-free finitely generated nilpotent group Γ such that the local Euler factors ζ

scholarly journals IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

2020 ◽  
Vol 71 (3) ◽  
pp. 959-980
Author(s):  
Christopher Voll
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Infinite Family ◽  
Nilpotent Groups ◽  
Unipotent Group ◽  
Group Schemes ◽  
Lie Rings ◽  
Local Functional ◽  
Explicit Formulae ◽  
Local Functional Equations

Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

The zeta function of [sfr ][lfr ]2 and resolution of singularities

2002 ◽  
Vol 132 (1) ◽  
pp. 57-73 ◽  
Author(s):  
MARCUS DU SAUTOY ◽  
GARETH TAYLOR
Keyword(s):  
Zeta Function ◽  
Finite Index ◽  
General Result ◽  
Quantifier Elimination ◽  
Rational Functions ◽  
Resolution Of Singularities ◽  
Euler Product ◽  
Product Decomposition ◽  
Local Factors ◽  
Local Zeta Function

Let L be a ring additively isomorphic to ℤd. The zeta function of L is defined to bewhere the sum is taken over all subalgebras H of finite index in L. This zeta function has a natural Euler product decomposition:These functions were introduced in a paper of Grunewald, Segal and Smith [5] where the local factors ζL[otimes ]ℤp(s) were shown to always be rational functions in p−s. The proof depends on representing the local zeta function as a definable p-adic integral and then appealing to a general result of Denef’s [1] about the rationality of such integrals. The proof of Denef relies on Macintyre’s Quantifier Elimination for ℚp [8] followed by techniques developed by Igusa [6] which employ resolution of singularities.

scholarly journals REMARKS ON THE MIXED JOINT UNIVERSALITY FOR A CLASS OF ZETA FUNCTIONS

2016 ◽  
Vol 95 (2) ◽  
pp. 187-198 ◽  
Author(s):  
ROMA KAČINSKAITĖ ◽  
KOHJI MATSUMOTO
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Functional Independence ◽  
The Other ◽  
Hurwitz Zeta Function ◽  
Euler Product ◽  
Joint Universality

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

ON THE EULER PRODUCT OF THE DEDEKIND ZETA FUNCTION

2009 ◽  
Vol 05 (02) ◽  
pp. 293-301
Author(s):  
XIAN-JIN LI
Keyword(s):  
Zeta Function ◽  
Algebraic Number ◽  
Zeta Functions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Product Formula ◽  
Euler Product ◽  
Dedekind Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζk(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

scholarly journals RUELLE TYPE L-FUNCTIONS VERSUS DETERMINANTS OF LAPLACIANS FOR TORSION FREE ABELIAN GROUPS

2008 ◽  
Vol 19 (08) ◽  
pp. 957-979
Author(s):  
NOBUSHIGE KUROKAWA ◽  
MASATO WAKAYAMA ◽  
YOSHINORI YAMASAKI
Keyword(s):  
Zeta Function ◽  
Imaginary Axis ◽  
Free Abelian Group ◽  
Arithmetic Function ◽  
Euler Product ◽  
Natural Boundary ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Determinants Of Laplacians ◽  
Torsion Free

We study Ruelle's type zeta and L-functions for a torsion free abelian group Γ of rank ν ≥ 2 defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when ν = 2, 4 and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus ℝν/Γ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

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