论文标题
有趣而有用的兰伯特系列身份的目录
A catalog of interesting and useful Lambert series identities
论文作者
论文摘要
Lambert系列生成函数是一个特殊的系列，该系列汇总在\ [l_f（q）定义的算术函数$ f $上：= \ sum_ {n \ geq 1} \ frac {f（n）q^n} {1-q^n} = \ sum_ = \ sum_ {m \ geq 1}（m \ geq 1}（m） \]由于这种类型的生成功能的左侧术语的方式产生了Dirichlet卷积卷积的$ f $的除数总和，因此这些扩展是枚举数字理论中许多乘法特殊功能的普通生成函数的自然方法。我们概述了兰伯特系列生成功能扩展的关键特性，它们的组合概括，并包括了表格的表格，其中说明了这些系列特殊情况的已知公式。从这个意义上讲，我们更多地专注于兰伯特系列列举的序列的形式属性，并且不花大量时间将这些系列作为受严格合并约束的分析对象。 在阅读此文档之前，可能会问的第一个问题是：为什么汇编了有趣的兰伯特系列身份目录？与H. W. Gould和T. Shonhiwa的必不可少的参考一样，有趣的Dirichlet系列目录，用于Dirichlet系列（DGF）身份，在许多情况下，人们需要在Lambert系列及其性质上进行摘要参考。最近，通过扩展其生成功能将兰伯特系列的扩展与分区功能相关联。除了这些新的扩展并提供了兰伯特系列的介绍之外，我们还列出了兰伯特系列求和的经典相关和“赔率和结局”示例，这些示例偶尔在应用程序中很有用。
A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side terms of this type of generating function generate divisor sums of $f$ convolved by Dirichlet convolution with one, these expansions are natural ways to enumerate the ordinary generating functions of many multiplicative special functions in number theory. We present an overview of key properties of Lambert series generating function expansions, their more combinatorial generalizations, and include a compendia of tables illustrating known formulas for special cases of these series. In this sense, we focus more on the formal properties of the sequences that are enumerated by the Lambert series, and do not spend significant time treating these series as analytic objects subject to rigorous convergence constraints. The first question one might ask before reading this document is: Why has is catalog of interesting Lambert series identities compiled? As with the indispensible reference by H. W. Gould and T. Shonhiwa, A catalog of interesting Dirichlet series, for Dirichlet series (DGF) identities, there are many situations in which one needs a summary reference on Lambert series and their properties. New work has been done recently tying Lambert series expansions to partition functions by expansions of their generating functions. In addition to these new expansions and providing an introduction to Lambert series, we have listings of classically relevant and "odds and ends'' examples for Lambert series summations that are occasionally useful in applications. If you see any topics or identities the author has missed, please contact us over email to append to this reference.
