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拉馬努金和

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數學的分支領域數論中,拉馬努金和英语:Ramanujan's sum)常標示為cq(n),為一個帶有兩正整數變數q以及n 的函數,其定義如下:

其中(a, q) = 1表示a只能是與q互質的數。

斯里尼瓦瑟·拉馬努金於1918年的一篇論文中引入這項和的觀念。[1]拉馬努金和也用在維諾格拉多夫定理英语Vinogradov's theorem的證明,此定理指出:任何足夠大的奇數可為三個質數的和。[2]

本文符號彙整[编辑]

若整數ab,有關係(唸作「a整除b」),表示存在一個整數c使得b = ac;相似地,表示「a無法整除b」。

求和符號

表示d只採用其正整數因數m,亦即

另外用到的有:

cq(n)的數學式[编辑]

三角函數[编辑]

下面的式子源自於定義、歐拉公式以及基本三角函數恆等式:

等等(A000012, A033999, A099837, A176742,.., A100051, ...)。這些式子顯示出cq(n)為實數

拉馬努金展開式[编辑]

參考文獻[编辑]

  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

書目[编辑]

  • 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