Functional equations for poly-dimensional zeta functions and the evaluation of Madelung constants

1976 ◽  
Vol 9 (4) ◽  
pp. 499-505 ◽  
Author(s):  
I J Zucker
Keyword(s):  
Functional Equations ◽  
Zeta Functions

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.

Stability results for local zeta functions of groups algebras, and modules

2017 ◽  
Vol 165 (3) ◽  
pp. 435-444 ◽  
Author(s):  
TOBIAS ROSSMANN
Keyword(s):  
Group Theory ◽  
Functional Equations ◽  
Zeta Functions ◽  
Local Zeta Functions ◽  
Single Formula ◽  
Almost All ◽  
Stability Results ◽  
Zeta Functions Of Groups ◽  
Local Base

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

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