This is a very basic and pedagogical review of the concepts of zeta function and of the associated zeta regularization method, starting from the notions of harmonic series and of divergent sums in general. By way of very simple examples, it is shown how these powerful methods are used for the regularization of physical quantities, such as quantum vacuum fluctuations in various contexts. In special, in Casimir effect setups, with a note on the dynamical Casimir effect, and mainly concerning its application in quantum theories in curved spaces, subsequently used in gravity theories and cosmology. The second part of this work starts with an essential introduction to large scale cosmology, in search of the observational foundations of the Friedmann-Lemaître-Robertson-Walker (FLRW) model, and the cosmological constant issue, with the very hard problems associated with it. In short, a concise summary of all these interrelated subjects and applications, involving zeta functions and the cosmos, and an updated list of the pioneering and more influential works (according to Google Scholar citation counts) published on all these matters to date, are provided.

中文翻译：

---

Zeta函数与宇宙-基本简要回顾

这是对zeta函数和相关的zeta正则化方法的概念的非常基础和教学性的回顾，从泛泛序列和和和的概念开始。通过非常简单的示例，显示了如何使用这些强大的方法对物理量进行正则化，例如在各种情况下进行量子真空波动。特别地，在卡西米尔效应装置中，对动力学卡西米尔效应作了说明，主要涉及其在弯曲空间量子理论中的应用，随后在重力理论和宇宙学中使用。这项工作的第二部分首先对大规模宇宙学进行必要的介绍，以寻找Friedmann-Lemaître-Robertson-Walker（FLRW）模型的观测基础以及宇宙学常数问题，与之相关的非常棘手的问题。简而言之，本文提供了所有这些相互关联的主题和应用的简要摘要，其中涉及zeta函数和宇宙，并提供了迄今为止针对所有这些问题出版的具有开创性和影响力的作品（根据Google Scholar引用计数）的更新列表。 。