拉馬努金和

在數學的分支領域數論中，拉馬努金和（英語：Ramanujan's sum）常標示為${\displaystyle c_{q}(n)}$，為一個帶有兩正整數變量${\displaystyle q}$以及${\displaystyle n}$的函數，其定義如下：

${\displaystyle c_{q}(n)=\sum _{a=1 \atop (a,q)=1}^{q}e^{2\pi i{\tfrac {a}{q}}n},}$

提示：此條目頁的主題不是拉馬努金求和。

其中${\displaystyle (a,q)=1}$表示${\displaystyle a}$只能是與${\displaystyle q}$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（英語：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

若整數a與b，有關係${\displaystyle a\mid b}$（唸作「a整除b」），表示存在一個整數c使得b = ac；相似地，${\displaystyle a\nmid b}$表示「a無法整除b」。

求和符號

${\displaystyle \sum _{d\,\mid \,m}f(d)}$

表示d只採用其正整數因數m，亦即

${\displaystyle \sum _{d\,\mid \,12}f(d)=f(1)+f(2)+f(3)+f(4)+f(6)+f(12)}$。

另外用到的有：

• ${\displaystyle (a,\,b)\;}$為最大公因數，
• ${\displaystyle \phi (n)\;}$為歐拉總計函數，
• ${\displaystyle \mu (n)\;}$為莫比烏斯函數，以及
• ${\displaystyle \zeta (s)\;}$為黎曼ζ函數。

cq(n)的數學式

三角函數

下面的式子源自於定義、歐拉公式${\displaystyle e^{ix}=\cos x+i\sin x}$以及基本三角函數恆等式：

${\displaystyle {\begin{aligned}c_{1}(n)&=1\\c_{2}(n)&=\cos n\pi \\c_{3}(n)&=2\cos {\tfrac {2}{3}}n\pi \\c_{4}(n)&=2\cos {\tfrac {1}{2}}n\pi \\c_{5}(n)&=2\cos {\tfrac {2}{5}}n\pi +2\cos {\tfrac {4}{5}}n\pi \\c_{6}(n)&=2\cos {\tfrac {1}{3}}n\pi \\c_{7}(n)&=2\cos {\tfrac {2}{7}}n\pi +2\cos {\tfrac {4}{7}}n\pi +2\cos {\tfrac {6}{7}}n\pi \\c_{8}(n)&=2\cos {\tfrac {1}{4}}n\pi +2\cos {\tfrac {3}{4}}n\pi \\c_{9}(n)&=2\cos {\tfrac {2}{9}}n\pi +2\cos {\tfrac {4}{9}}n\pi +2\cos {\tfrac {8}{9}}n\pi \\c_{10}(n)&=2\cos {\tfrac {1}{5}}n\pi +2\cos {\tfrac {3}{5}}n\pi \\\end{aligned}}}$

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

拉馬努金展開式

參考文獻

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

• 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

• Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown（英語：Roger Heath-Brown） and J. H. Silverman（英語：Joseph H. Silverman）. Foreword by Andrew Wiles. 6th, Oxford: Oxford University Press, 2008 [1938], ISBN 978-0-19-921986-5, Zbl 1159.11001

• Knopfmacher, John, Abstract Analytic Number Theory 2nd, New York: Dover, 1990 [1975], ISBN 0-486-66344-2, Zbl 0743.11002

• Nathanson, Melvyn B., Additive Number Theory: the Classical Bases, Graduate Texts in Mathematics 164, Springer-Verlag, Section A.7, 1996, ISBN 0-387-94656-X, Zbl 0859.11002.
• Nicol, C. A. Some formulas involving Ramanujan sums. Canad. J. Math. 1962, 14: 284–286. doi:10.4153/CJM-1962-019-8.

• 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)

• Ramanujan, Srinivasa, Collected Papers, Providence RI: AMS / Chelsea, 2000, ISBN 978-0-8218-2076-6

• 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