Values of Zeta Functions and Their Applications
Values of Zeta Functions and Their Applications
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DOI：
10.1007/978-3-0348-9112-7_23
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
D. Zagier
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文献类型：
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作者：
D. Zagier
各种 Zeta 函数是现代数论中无处不在的对象，一个不断重复出现的主题是它们在积分参数中的特殊值所扮演的角色，这些特殊值以神秘的方式与底层几何联系在一起，并且通常似乎决定了与 Zeta 函数相关的对象的最重要属性。标题中的“应用程序”一词指的是后一个属性。在本文中，我们将对其中一些“应用”进行高度特殊和偏见的游览，不试图系统化，而只是为了让大家了解 zeta 函数的特殊值与其他有趣的数学问题相互关联的一些方式。典型的 zeta 函数是“黎曼”（数学），特殊值的典型结果是定理 z(k) = 有理数 × πk(k> 0 Even)，(1) 欧拉在 1735 年证明了该定理，我们将在第 1 节中对其进行简短证明。（本例中的“应用”是出现在该公式右侧的有理数在分圆域理论中所扮演的角色，在 p-adic zeta 函数的构造中，以及在费马大定理的研究中。）
Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem thatζ(k) = rational number × πk(k> 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction ofp-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)