拉马努金和

数学的分支领域数论中,拉马努金和(英语:Ramanujan's sum)常标示为,为一个带有两正整数变数以及的函数,其定义如下:

其中表示只能是与互质的数。

斯里尼瓦瑟·拉马努金于1918年的一篇论文中引入这项和的观念。[1]拉马努金和也用在维诺格拉多夫定理英语Vinogradov's theorem的证明,此定理指出:任何足够大的奇数可为三个质数的和。[2]

本文符号汇整

整数ab,有关系 (念作“a整除b”),表示存在一个整数c使得b = ac;相似地, 表示“a无法整除b”。

求和符号

 

表示d只采用其正整数因数m,亦即

 

另外用到的有:

cq(n)的数学式

三角函数

下面的式子源自于定义、欧拉公式 以及基本三角函数恒等式:

 

等等(A000012, A033999, A099837, A176742,.., A100051, ...)。这些式子显示出cq(n)为实数

拉马努金展开式

参考文献

  1. ^ Ramanujan, On Certain Trigonometric Sums ...

    These sums are obviously of great interest, and a few of their properties have been discussed already. But, so far as I know, they have never been considered from the point of view which I adopt in this paper; and I believe that all the results which it contains are new.

    (Papers, p. 179). In a footnote cites pp. 360–370 of the Dirichlet-Dedekind Vorlesungen über Zahlentheorie, 4th ed.
  2. ^ Nathanson, ch. 8

书目

  • Hardy, G. H., Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Providence RI: AMS / Chelsea, 1999, ISBN 978-0-8218-2023-0 
  • Ramanujan, Srinivasa, On Certain Trigonometric Sums and their Applications in the Theory of Numbers, Transactions of the Cambridge Philosophical Society, 1918, 22 (15): 259–276  (pp. 179–199 of his Collected Papers)
  • Ramanujan, Srinivasa, On Certain Arithmetical Functions, Transactions of the Cambridge Philosophical Society, 1916, 22 (9): 159–184  (pp. 136–163 of his Collected Papers)
  • Schwarz, Wolfgang; Spilker, Jürgen, Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties, London Mathematical Society Lecture Note Series 184, Cambridge University Press, 1994, ISBN 0-521-42725-8, Zbl 0807.11001