广义高阶Herglotz函数I.函数方程
Extended higher Herglotz functions I. Functional equations
摘要：
在1975年，唐Zagier获得了一个新版本的克罗内克极限公式的一个真正的二次领域，其中涉及一个有趣的功能$F(x)$这是现在已知的\emph{Herglotz function}。正如Zagier以及最近Radchenko和Zagier所证明的那样，$F(x)$满足在代数数论和解析数论中都感兴趣的美丽性质。在本文中，我们研究$\mathscr{F}_{k,N}(x)$，扩展的Herglotz功能，其中也包括\emph{higher Herglotz function}的Vlasenko和Zagier。我们称之为\emph{extended higher Herglotz function}。它与某个广义朗伯级数密切相关。本文导出了$\mathscr{F}_{k,N}(x)$满足的两类不同的函数方程。拉琴科和扎吉尔了一个美丽的关系之间的积分$\displaystyle\int_{0}^{1}\frac{\log(1+t^x)}{1+t}\, dt$和$F(x)$，并用它来评估这一积分在各种合理的以及非理性的论点。我们得到了$\mathscr{F}_{k,N}(x)$和上述积分的一个推广的关系，涉及多对数。得到了$\mathscr{F}_{k, N}(x)$和某些广义Lambert级数的渐近展开式，并得到了其它一些补充结果.
英文摘要：
In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function $F(x)$ which is now known as the \emph{Herglotz function}. As demonstrated by Zagier, and very recently by Radchenko and Zagier, $F(x)$ satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study $\mathscr{F}_{k,N}(x)$, an extension of the Herglotz function which also subsumes \emph{higher Herglotz function} of Vlasenko and Zagier. We call it the \emph{extended higher Herglotz function}. It is intimately connected with a certain generalized Lambert series. We derive two different kinds of functional equations satisfied by $\mathscr{F}_{k,N}(x)$. Radchenko and Zagier gave a beautiful relation between the integral $\displaystyle\int_{0}^{1}\frac{\log(1+t^x)}{1+t}\, dt$ and $F(x)$ and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between $\mathscr{F}_{k,N}(x)$ and a generalization of the above integral involving polylogarithm. The asymptotic expansions of $\mathscr{F}_{k, N}(x)$ and some generalized Lambert series are also obtained along with other supplementary results.
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