On the Euler Product of Some Zeta Functions

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.

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000733 ◽

2016 ◽

Vol 95 (2) ◽

pp. 187-198 ◽

Cited By ~ 2

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.

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ZETA FUNCTIONS OF GROUPS: EULER PRODUCTS AND SOLUBLE GROUPS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500000456 ◽

2002 ◽

Vol 45 (1) ◽

pp. 149-154

Author(s):

Marcus du Sautoy

Keyword(s):

Zeta Functions ◽

Nilpotent Groups ◽

Subject Classification ◽

Euler Product ◽

Soluble Groups ◽

Euler Products ◽

Primary 20F16 ◽

Zeta Functions Of Groups

AbstractThe well-behaved Sylow theory for soluble groups is exploited to prove an Euler product for zeta functions counting certain subgroups in pro-soluble groups. This generalizes a result of Grunewald, Segal and Smith for nilpotent groups.AMS 2000 Mathematics subject classification: Primary 20F16; 11M99

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ON THE EULER PRODUCT OF THE DEDEKIND ZETA FUNCTION

International Journal of Number Theory ◽

10.1142/s1793042109002109 ◽

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.

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EULER PRODUCT EXPRESSION OF TRIPLE ZETA FUNCTIONS

International Journal of Mathematics ◽

10.1142/s0129167x05002825 ◽

2005 ◽

Vol 16 (02) ◽

pp. 111-136

Author(s):

HIROTAKA AKATSUKA

Keyword(s):

Tensor Products ◽

Zeta Functions ◽

Summation Formula ◽

Poisson Summation Formula ◽

Euler Product ◽

Multiple Zeta ◽

Product Expression ◽

Triple Zeta ◽

Sine Functions ◽

Poisson Summation

We construct multiple zeta functions considered as absolute tensor products of usual zeta functions. We establish Euler product expressions for triple zeta functions [Formula: see text] with p, q, r distinct primes, via multiple sine functions by using the signatured Poisson summation formula. We also establish Euler product expressions for triple zeta functions [Formula: see text] with a prime p, via the theory of multiple sine functions.

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