# Zeta functions of groups and rings

@inproceedings{Sautoy2007ZetaFO, title={Zeta functions of groups and rings}, author={Marcus du Sautoy and Fritz Grunewald}, year={2007} }

We report on progress and problems concerning the analytical behaviour of the zeta
functions of groups and rings. We also describe how these generating functions are special cases
of adelic cone integrals for which our results hold.

## 76 Citations

Zeta functions of groups and rings - recent developments

- Mathematics
- 2014

I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or…

A Framework for Computing Zeta Functions of Groups, Algebras, and Modules

- Mathematics
- 2017

We give an overview of the author’s recent work on methods for explicitly computing various types of zeta functions associated with algebraic counting problems. Among the types of zeta functions that…

Essential singularities of Euler products

- Mathematics
- 2009

We classify singularities of Dirichlet series having Euler products which are rational functions for p and p^{-s} for p a prime number and give examples of natural boundaries from zeta functions of…

Analytic Continuation of some zeta functions

- Mathematics
- 2010

This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach…

A remark on the degree of polynomial subgroup growth of nilpotent groups

- Mathematics
- 2021

We show that if two finitely generated nilpotent groups have isomorphic C-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. A similar…

Computing local zeta functions of groups, algebras, and modules

- Mathematics
- 2016

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local…

Representation growth and representation zeta functions of groups

- Mathematics
- 2012

We give a short introduction to the subject of representation growth and representation zeta functions of groups, omitting all proofs. Our focus is on results which are relevant to the study of…

Reduced zeta functions of Lie algebras

- Mathematics
- 2007

Abstract We define reduced zeta functions of Lie algebras, which can be derived, via the Euler characteristic, from motivic zeta functions counting subalgebras and ideals. We show that reduced zeta…

Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

- Mathematics
- 2015

We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power…

## References

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- Mathematics
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We use the theory of zeta functions of groups to establish a lower limit for the degree of polynomial normal subgroup growth in class two nilpotent groups.

The Normal Zeta Function of the Free Class Two Nilpotent Group on Four Generators

- Mathematics
- 2005

We calculate explicitly the normal zeta function of the free group of class two on four generators, denoted by F2,4. This has Hirsch length ten.

Normal subgroup growth in free class-2-nilpotent groups

- Mathematics
- 2004

Abstract.Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions prove that they satisfy local functional equations and determine their abscissae of…

Zeta Functions of Groups

- Mathematics
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A zeta function is an analytic function whose analytic properties somehow encapsulate a tremendous amount of arithmetic information. These functions are an uncannily powerful tool in number theory;…

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Zeta functions related to the group SL2(ℤp)

- Mathematics
- 1999

An explicit formula is given for the number of subgroups of indexpn in the principle congruence subgroups of SL2(ℤp) (for odd primesp), and for the zeta function associated with the group.…

Zeta Functions of Crystallographic Groups and Analytic Continuation

- Mathematics
- 1999

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical…

Functional equations for local normal zeta functions of nilpotent groups

- Mathematics
- 2003

Abstract.We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically…

Computations on Nilpotent Lie Algebras of Dimension 6

- Mathematics
- 2009

Centers, regular elements and free nilpotent generators of all nilpotent Lie algebras of dimension 6 has been calculated and listed in tables.