论文标题
在某些Zeta积分上：转换公式
On certain zeta integral: Transformation formula
论文作者
论文摘要
我们引入了一个“ $ l $ function” $ \ MATHCAL {l} $从Barnes的多个Zeta函数$ζ$的积分表示构建的。与后者不同，$ \ Mathcal {l} $是在配备了组$ g $的非平凡动作的域上定义的。尽管这两个功能彼此不同，但我们可以使用$ \ Mathcal {l} $来研究$ζ$。实际上，在$ g $转换下的$ \ Mathcal {l} $的转换公式为我们提供了有关$ζ$及其$ s $衍生的特殊值的新观点。特别是，当限制在$ g $的元素固定的点时，我们将获得$ζ$的kronecker限制公式。为了说明这一原则，我们评估了统一根部的某些广义兰伯特系列，从而确立了相关的代数结果。此外，我们在统一的根部作为某种无限产品表达了Barnes的多个伽马功能。应该提到的是，这项工作还考虑了$ζ$的扭曲版本。
We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $ζ$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $ζ$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $ζ$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $ζ$ when restricted to points fixed by elements of $G$. As an illustration of this principle, we evaluate certain generalized Lambert series at roots of unity, establishing pertinent algebraicity results. Also, we express the Barnes' multiple gamma function at roots of unity as a certain infinite product. It should be mentioned that this work also considers twisted versions of $ζ$.
