# 拉馬努金和

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

## 本文符號彙整

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

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

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

## cq(n)的數學式

### 三角函數

{\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}}}

## 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
• 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