Riemann zeta functions for Krull monoids

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with...

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Bibliographic Details
Published inJournal of number theory Vol. 262; pp. 134 - 160
Main Authors Gotti, Felix, Krause, Ulrich
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.09.2024
Subjects
Decay theorem
Euler's product
Krull monoid
Prime
Riemann zeta function
Strong atom
secondary
Strong atom
Euler's product
Prime
Decay theorem
Riemann zeta function
Krull monoid
primary
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Summary:The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2024.03.001