拉马努金和
在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量以及的函数,其定义如下:
其中表示只能是与互素的数。
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的证明,此定理指出:任何足够大的奇数可为三个素数的和。[2]
本文符号汇整
若整数a与b,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只采用其正整数约数m,亦即
- 。
另外用到的有:
cq(n)的数学式
三角函数
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
- ^ Ramanujan, On Certain Trigonometric Sums ...
(Papers, p. 179). In a footnote cites pp. 360–370 of the Dirichlet-Dedekind Vorlesungen über Zahlentheorie, 4th ed.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.
- ^ 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
- Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. 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