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 c q ( n ) {\displaystyle c_{q}(n)} ，為一個帶有兩正整數變數 q {\displaystyle q} 以及 n {\displaystyle n} 的函數，其定義如下： c q ( n ) = ∑ a = 1 ( a , q ) = 1 q e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{a=1 \atop (a,q)=1}^{q}e^{2\pi i{\tfrac {a}{q}}n},} 其中 ( a , q ) = 1 {\displaystyle (a,q)=1} 表示 a {\displaystyle a} 只能是與 q {\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。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。 若整數a與b，有關係 a ∣ b {\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， a ∤ b {\displaystyle a\nmid b} 表示「a無法整除b」。 求和符號 ∑ d ∣ m f ( d ) {\displaystyle \sum _{d\,\mid \,m}f(d)} 表示d只採用其正整數因數m，亦即 ∑ d ∣ 12 f ( d ) = f ( 1 ) + f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 6 ) + f ( 12 ) {\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： ( a , b ) {\displaystyle (a,\,b)\;} 為最大公因數， ϕ ( n ) {\displaystyle \phi (n)\;} 為歐拉總計函數， μ ( n ) {\displaystyle \mu (n)\;} 為莫比烏斯函數，以及 ζ ( s ) {\displaystyle \zeta (s)\;} 為黎曼ζ函數。 下面的式子源自於定義、歐拉公式 e i x = cos ⁡ x + i sin ⁡ x {\displaystyle e^{ix}=\cos x+i\sin x} 以及基本三角函數恆等式： c 1 ( n ) = 1 c 2 ( n ) = cos ⁡ n π c 3 ( n ) = 2 cos ⁡ 2 3 n π c 4 ( n ) = 2 cos ⁡ 1 2 n π c 5 ( n ) = 2 cos ⁡ 2 5 n π + 2 cos ⁡ 4 5 n π c 6 ( n ) = 2 cos ⁡ 1 3 n π c 7 ( n ) = 2 cos ⁡ 2 7 n π + 2 cos ⁡ 4 7 n π + 2 cos ⁡ 6 7 n π c 8 ( n ) = 2 cos ⁡ 1 4 n π + 2 cos ⁡ 3 4 n π c 9 ( n ) = 2 cos ⁡ 2 9 n π + 2 cos ⁡ 4 9 n π + 2 cos ⁡ 8 9 n π c 10 ( n ) = 2 cos ⁡ 1 5 n π + 2 cos ⁡ 3 5 n π {\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cq(n)為實數。 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. Nathanson, ch. 8 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（英语：Roger Heath-Brown） and J. H. Silverman（英语：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