论文标题
Herglotz功能的算术特性
Arithmetic properties of the Herglotz function
论文作者
论文摘要
在本文中，我们研究了两个函数$ f（x）$和$ j（x）$，最初是由赫格洛茨（Herglotz）在1923年发现的，后来又被一位作者重新发现并使用了与Kronecker限制限制公式有关的真实二次领域。我们讨论了这些功能的许多有趣的属性，包括在理性或二次非理性参数上作为对数的二合作和乘积的合理线性组合，来自Hecke操作员的功能方程以及与Stark的猜想的联系。我们还与模块化组$ \ mathrm {psl}（2，\ Mathbb {z}）$的1个环节讨论了连接。
In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group $\mathrm{PSL}(2,\mathbb{Z})$.
