Some examples of generalized zeta regularized products
\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 \[ \dotprod_{n \in I}a_n=\exp\left(-\text{Res}_{s=0}\frac{\zeta_{\mathbf{a}}(s)}{s^2}\right). \] If \(\zeta_{\mathbf{a}}(s)\) holomorphic at \(s=0\), then \(\regprod_{n \in I}a_n=\dotprod_{n \in I}a_n\). In the present paper, the authors give various examples of calculations using the generalized zeta regularized products, also they study properties of the functions defined by the zeta regularized products. We will mention some examples. Denote by \(m(n)\) the multiplicity of \(n\) in the sequence \({\mathbf{a}}\). Then \[ \zeta_{\mathbf{a}}(s)=\sum_{m=1}^{\infty}\frac{m(n)}{n^s}, \] and the notation \(\regprod_{n \in I}a_n=\regprod_{n=1}^{\infty}(n;m(n))\) well as \(\dotprod_{n \in I}a_n=\dotprod_{n=1}^{\infty}(n;m(n))\) can be used. For example, \[ \regprod_{n=1}^{\infty}(n;n^k)=\exp\{-\zeta'(-k)\}, \] where \(\zeta(s)\) is the Riemann zeta-function. Moreover, \[ \regprod_{n: \;\text{ square-free}}n=\regprod_{n: \;\text{ square}}n=2 \pi. \] However, \(\regprod_{n: \;\text{ square-free}}(n;n)\) can't be calculated because the attached zeta-function \[ \sum_{n: \;\text{ square-free}} \frac{n}{n^s}=\frac{\zeta(s-1)}{\zeta(2s-2)} \] is note holomorphic at \(s=0\). Therefore, the authors use the generalized notion of the zeta regularized product, and prove that \[ \dotprod_{n: \;\text{ square-free}}(n;n) = \exp(-\frac{1}{4}\frac{\zeta''(-1)}{\zeta'(-2)}+\frac{1}{2} \frac{\zeta'(-1)\zeta''(-2)}{\zeta'(-2)^2} -\frac{1}{36}\frac{\zeta'''(-2)}{\zeta'(-2)^2}+\frac{1}{24} \frac{\zeta''(-2)^2}{\zeta'(-2)^3}). \] Now some examples of functions defined by using the zeta regularized products. Let \(q>1\), then \[ \dotprod_{n=0}^{\infty}q^{n+x}=q^{-\frac{1}{2}B_2(x)}, \] where \(B_2(x)\) is the Bernoulli polynomial of degree 2. Let \(\gamma\) and \(\gamma_1\) be defined by \[ \zeta(s)=\frac{1}{s-1}+\gamma+\gamma_1(s-1)+O(s^2). \] Then \[ \dotprod_{m,n \geq 1}(mn-x)=\frac{1}{\sqrt{2 \pi}}e^{-(\gamma^2+2\gamma_1)x}\prod_{m,n \geq 1} (1-\frac{x}{m n})e^{\frac{x}{mn}}. \] In general, the equality \[ \dotprod_{n \in I}a_n \dotprod_{n \in I} b_n=\dotprod_{n \in I}a_n b_n \] is not true. The authors present examples when the later equality is true, and is not true. They also apply the properties of the zeta regularized products to prove properties of known functions, for example, the multiplication formula for the gamma-function.
scientific article; zbMATH DE number 2157863