Some examples of generalized zeta regularized products
type: Item
instance of: scholarly article
title: Some examples of generalized zeta regularized products
zbMATH Open document ID: 1090.11053
author: Kazufumi Kimoto
author: Nobushige Kurokawa
author: Chie Sonoki
author: Masato Wakayama
published in: Kodai Mathematical Journal
publication date: 2005-04-18T00:00:00Z
review text: \def\regprod{\mathop{ \ooalign{\(\prod\)\crcr\(\coprod\)}}} \def\Regprod{\mathop{ \ooalign{\(\displaystyle\prod\)\crcr\(\displaystyle\coprod\)}}\displaylimits} \def\dotprod{\mathop{ \ooalign{\(\prod\)\crcr\hfil\raise0.3ex \hbox{\(\scriptscriptstyle\bullet\)}\hfil\crcr\(\coprod\)}}} \def\Dotprod{\mathop{ \ooalign{\(\displaystyle\prod\)\crcr\hfil\raise0.2ex \hbox{\(\scriptstyle\bullet\)}\hfil\crcr\(\displaystyle\coprod\)}}\displaylimits} \def\ddotprod{\mathop{ \ooalign{\(\prod\)\crcr\hfil\raise0.75ex \hbox{\(\scriptscriptstyle\bullet\)}\hfil\crcr\hfil\lower0.2ex \hbox{\(\scriptscriptstyle\bullet\)}\hfil\crcr\(\coprod\)}}} \def\Ddotprod{\mathop{ \ooalign{\(\displaystyle\prod\)\crcr\hfil\raise0.8ex \hbox{\(\scriptstyle\bullet\)}\hfil\crcr\hfil\lower0.4ex \hbox{\(\scriptstyle\bullet\)}\hfil\crcr\(\displaystyle\coprod\)}}\displaylimits} For a given sequence \({\mathbf{a}}=\{a_n\}_{n \in I}\), the zeta-function \(\zeta_{\mathbf{a}}(s)\) attached to \({\mathbf{a}}\) is defined by \[ \zeta_{\mathbf a}(s)=\sum_{n \in I}a_n^{-s}. \] Suppose that the function \(\zeta_{\mathbf a}(s)\) is analytically continuable to some region containing the point \(s=0\) and is holomorphic at \(s=0\). Then the zeta regularized product of the sequence \({\mathbf{a}}\) is defined by \[ \regprod_{n \in I}a_n=\exp(-\zeta'_{{\mathbf{a}}}((0)). \] This notion was successfully applied by \textit{C. Deninger} [Invent. Math. 107, 135--150 (1992; Zbl 0762.14015)] and \textit{A. Voros} [Commun. Math. Phys. 110, 439--465 (1987; Zbl 0631.10025)] to study various zeta-functions. Recently Kurokawa and Wakayama extended the notion of the regularized product. If \(\zeta_{{\mathbf{a}}}(s)\) is analytically continuable to some region containing the point \(s=0\), and this point is its pole, then the sequence \({\mathbf{a}}\) is called regularizable, and the generalized zeta regularized product of \({\mathbf{a}}\) is defined by \[ \dotprod_{n \in I}a_n=\exp\left(-\text{Res}_{s=0}\frac{\zeta_{\mathbf{a}}(s)}{s^2}\r…
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 2157863
zbMATH Keywords: attached zeta function
zbMATH Keywords: generalized zeta regularization
zbMATH Keywords: zeta regularized product
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1104247354
OpenAlex ID: W2022838399
DOI: 10.2996/KMJ/1104247354
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description: scientific article; zbMATH DE number 2157863