欧拉乘积

假设${\displaystyle a}$为一积性函数，则狄利克雷级数

${\displaystyle \sum _{n}a(n)n^{-s}\,}$

等于欧拉乘积

${\displaystyle \prod _{p}P(p,s)\,}$

其中，乘积对所有素数${\displaystyle p}$进行，${\displaystyle P(p,s)}$则可表示为

${\displaystyle 1+a(p)p^{-s}+a(p^{2})p^{-2s}+\cdots .}$

这可以看作形式母函数，形式欧拉乘积展开的存在性与${\displaystyle a(n)}$为积性函数两者互为充要条件。

${\displaystyle a(n)}$为完全积性函数时可得到一重要的特例。此时${\displaystyle P(p,s)}$为等比级数，有

${\displaystyle P(p,s)={\frac {1}{1-a(p)p^{-s}}},}$

当${\displaystyle a(n)=1}$时即为黎曼ζ函数，更一般的情形则是狄利克雷特征。

• G. Polya, Induction and Analogy in Mathematics Volume 1 Princeton University Press (1954) L.C. Card 53-6388 (A very accessible English translation of Euler's memoir regarding this "Most Extraordinary Law of the Numbers" appears starting on page 91)
• Apostol, Tom M., , Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, 1976, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001 (Provides an introductory discussion of the Euler product in the context of classical number theory.)
• G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th ed., Oxford (1979) ISBN 0-19-853171-0 (Chapter 17 gives further examples.)
• George E. Andrews, Bruce C. Berndt, Ramanujan's Lost Notebook: Part I, Springer (2005), ISBN 0-387-25529-X
• G. Niklasch, Some number theoretical constants: 1000-digit values"