欧拉乘积

数论中,欧拉乘积英語:)是指狄利克雷级数可表示为一指标为素数无穷乘积。这一乘积以瑞士数学家莱昂哈德·欧拉的名字命名,他证明了黎曼ζ函数可表示为此无穷乘积的形式。

定义

假设为一积性函数,则狄利克雷级数

等于欧拉乘积

其中,乘积对所有素数进行,则可表示为

这可以看作形式母函数,形式欧拉乘积展开的存在性与为积性函数两者互为充要条件。

完全积性函数时可得到一重要的特例。此时等比级数,有

时即为黎曼ζ函数,更一般的情形则是狄利克雷特征

参考文献

  • 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"
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