查看“︁拉馬努金和”︁的源代码

{{NoteTA|G1=Math}}
{{distinguish|拉馬努金求和}}
在[[數學]]的分支領域[[數論]]中，'''拉馬努金和'''（{{lang-en|Ramanujan's sum}}）常標示為<math>c_q(n)</math>，為一個帶有兩[[正整數]]變數<math>q</math>以及<math>n</math>的函數，其定義如下：
:<math>c_q(n)= \sum_{a=1\atop (a,q)=1}^q e^{2 \pi i \tfrac{a}{q} n},</math>

其中<math>(a,q)=1</math>表示<math>a</math>只能是與<math>q</math>[[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。<ref>Ramanujan, ''On Certain Trigonometric Sums ...'' <blockquote>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.</blockquote>(''Papers'', p. 179). In a footnote cites pp. 360–370 of the Dirichlet-Dedekind ''Vorlesungen über Zahlentheorie'', 4th ed.</ref>拉馬努金和也用在{{le|維諾格拉多夫定理|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。<ref>Nathanson, ch. 8</ref>

== 本文符號彙整 ==
若[[整數]]''a''與''b''，有關係<math>a\mid b</math>（唸作「''a''整除''b''」），表示存在一個整數''c''使得''b'' = ''ac''；相似地，<math>a\nmid b</math>表示「''a''無法整除''b''」。

求和符號
:<math>\sum_{d\,\mid\,m}f(d)</math>
表示''d''只採用其正整數[[因數]]''m''，亦即
:<math>\sum_{d\,\mid\,12}f(d) = f(1) + f(2) + f(3) + f(4) + f(6) + f(12) </math>。

另外用到的有：
* <math>(a,\,b)\;</math>為[[最大公因數]]，
* <math>\phi(n)\;</math>為[[歐拉總計函數]]，
* <math>\mu(n)\;</math>為[[莫比烏斯函數]]，以及
* <math>\zeta(s)\;</math>為[[黎曼ζ函數]]。

== ''c''<sub>''q''</sub>(''n'')的數學式 ==
===三角函數===
下面的式子源自於定義、[[歐拉公式]]<math>e^{ix}= \cos x + i \sin x</math>以及基本[[三角函數]]恆等式：
:<math>\begin{align}
c_1(n) &= 1 \\
c_2(n) &= \cos n\pi \\
c_3(n) &= 2\cos \tfrac23 n\pi \\
c_4(n) &= 2\cos \tfrac12 n\pi \\
c_5(n) &= 2\cos \tfrac25 n\pi + 2\cos \tfrac45 n\pi \\
c_6(n) &= 2\cos \tfrac13 n\pi \\
c_7(n) &= 2\cos \tfrac27 n\pi + 2\cos \tfrac47 n\pi + 2\cos \tfrac67 n\pi \\
c_8(n) &= 2\cos \tfrac14 n\pi + 2\cos \tfrac34 n\pi \\
c_9(n) &= 2\cos \tfrac29 n\pi + 2\cos \tfrac49 n\pi + 2\cos \tfrac89 n\pi \\
c_{10}(n)&= 2\cos \tfrac15 n\pi + 2\cos \tfrac35 n\pi \\
\end{align}</math>

等等（{{OEIS2C|A000012}}, {{OEIS2C|A033999}}, {{OEIS2C|A099837}}, {{OEIS2C|A176742}},.., {{OEIS2C|A100051}}, ...）。這些式子顯示出''c<sub>q</sub>''(''n'')為[[實數]]。

== 拉馬努金展開式 ==

== 參考文獻 ==
{{reflist}}
===書目===
*{{citation
  | last1 = Hardy  | first1 = G. H.
  | title = Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work
  | publisher = AMS / Chelsea
  | location = Providence RI
  | year = 1999
  | isbn = 978-0-8218-2023-0}}

*{{Citation
  | last1=Hardy
  | first1=G. H.
  | author1-link=G. H. Hardy
  | last2=Wright
  | first2=E. M.
  | author2-link=E. M. Wright
  | edition={{{edition|6th}}}
  | others={{#if: {{{edition|}}}||Revised by {{tsl|en|Roger Heath-Brown||D. R. Heath-Brown}} and {{tsl|en|Joseph H. Silverman||J. H. Silverman}}.  Foreword by [[安德魯·懷爾斯|Andrew Wiles]].}}
  | title=An Introduction to the Theory of Numbers
  | publisher={{#if: {{{edition|}}} | Clarendon Press | [[牛津大學出版社|Oxford University Press]] }}
  | location=Oxford
  | series=
  | isbn={{#switch: {{{edition|}}} | 4th = 0-19-853310-1 | 5th = 0-19-853171-0 | 978-0-19-921986-5 }}
  | mr=
  | zbl= {{#switch: {{{edition|}}} | 5th =  0423.10001 | 4th = 0086.25803 | 1159.11001 }}
  | year={{#switch: {{{edition|}}} | 4th = 1960 | 5th = 1979 | 2008 }}
  | origyear=1938}}

*{{citation
  | last1 = Knopfmacher | first1 = John
  | title = Abstract Analytic Number Theory
  | publisher = Dover | edition=2nd | zbl=0743.11002
  | location = New York
  | year = 1990 | origyear=1975
  | isbn = 0-486-66344-2}}

*{{citation
  | title=Additive Number Theory: the Classical Bases  
  | volume=164 
  | series=Graduate Texts in Mathematics 
  | last=Nathanson 
  | first=Melvyn B. 
  | publisher=Springer-Verlag 
  | year=1996 | isbn=0-387-94656-X | zbl= 0859.11002 | at= Section A.7 }}.
*{{cite journal | title=Some formulas involving Ramanujan sums | url=https://archive.org/details/sim_canadian-journal-of-mathematics_1962_14_2/page/284 |
  year=1962|journal=Canad. J. Math. | pages=284-286| doi=10.4153/CJM-1962-019-8
  |first1=C. A. | last1=Nicol |volume=14 }}

*{{citation
  | last1 = Ramanujan  | first1 = Srinivasa
  | title = On Certain Trigonometric Sums and their Applications in the Theory of Numbers 
  | journal = Transactions of the Cambridge Philosophical Society
  | volume = 22
  | issue = 15
  | year = 1918
  | pages = 259–276}} (pp.&nbsp;179–199 of his ''Collected Papers'')

*{{citation
  | last1 = Ramanujan  | first1 = Srinivasa
  | title = On Certain Arithmetical Functions
  | journal = Transactions of the Cambridge Philosophical Society
  | volume = 22
  | issue = 9
  | year = 1916
  | pages = 159–184}} (pp.&nbsp;136–163 of his ''Collected Papers'')

*{{citation
  | last1 = Ramanujan  | first1 = Srinivasa
  | title = Collected Papers
  | publisher = AMS / Chelsea
  | location = Providence RI
  | year = 2000
  | isbn = 978-0-8218-2076-6}}

* {{citation | first1=Wolfgang | last1=Schwarz | first2=Jürgen | last2=Spilker | title=Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties | year=1994 | publisher=[[劍橋大學出版社|Cambridge University Press]] | isbn=0-521-42725-8 | zbl=0807.11001 | series=London Mathematical Society Lecture Note Series | volume=184 }}


[[Category:数论]]
[[Category:数论中的平方]]
[[Category:斯里尼瓦瑟·拉马努金]]