Generalized Lereh formulas

The zeta-regularized product of a countable sequence $\{\lambda_{k}\}\subset \mathrm{C}\backslash \{0\} $ is defined by $\overline{\prod}_{k}\lambda_{k}=\exp(-\frac{\partial}{\partial s}\sum_{k}\lambda_{k}^{-s}|_{s=0}))$ provided that $\mathrm{A}(s)=\Sigma_{k}\lambda_{k}^{-s} $ is continued holomorphically at $s=0$. Here the branch is chosen so that $-\pi<\arg(\lambda_{k})\leq\pi $. There are several interesting formulas which can be formulated in terms of zeta-regularized products. Typical examples are Lerch’s formula $\overline{\prod}_{n=0}^{\infty}(n+x)=\frac{\sqrt{2\pi}}{\Gamma(x)} $ (1) and Kronecker’s limit formula $\overline{\prod}_{(\mathrm{c}_{J}d)=1}\frac{|cz+d|}{\sqrt{y}}=(y^{6}|\Delta(z)|)^{-\frac{1}{6}} $. (2) Here $\Gamma(x) $ is Euler’s gamma function and $\Delta(z)=e^{2\pi iz}\prod_{n=1}^{\infty}(1-e^{2\pi inz})^{2}4 $ is Ramanujan’s delta function. In this paper, we generalize Lerch’s formul