Dedekind sum

hbghlyj

$p,q$为不同的奇素数,定义Dedekind sum$$
s(p, q) \equiv \sum_{i=1}^{q}\left(\kern-5pt\left(\frac{i}{q}\right)\kern-5pt\right)\left(\kern-5pt\left(\frac{p i}{q}\right)\kern-5pt\right)
$$其中$$
(\!(x)\!) \equiv \begin{cases}x-\lfloor x\rfloor-\frac{1}{2} & x \notin \mathbb{Z} \\ 0 & x \in \mathbb{Z}\end{cases}
$$$[x]$是不超过$x$的最大整数. 求证$$
s(p, q)=\frac{1}{4 q} \sum_{r=1}^{q-1} \cot \left(\frac{\pi p r}{q}\right) \cot \left(\frac{\pi r}{q}\right)
$$

Czhang271828

See the third proof of THM 1, Ch 2 in this book. Therefore, we can define the Dedekind sum (of finite many pairwise coprime numbers) in higher dimension. See here.

hbghlyj

存档成功

isee

((x)) 是约定的，还是只是这里临时定义的？

hbghlyj

isee 发表于 2022-8-8 16:01
((x)) 是约定的，还是只是这里临时定义的？

Sawtooth function

${\displaystyle x(t)=t-\lfloor t\rfloor }$
$\displaystyle \hphantom{x(t)}=t\bmod 1\rule[20px]{0px}{0px}$
$\displaystyle \hphantom{x(t)}=-{\frac {2}{\pi }}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{k}}\sin \left(2\pi kt\right)$

hbghlyj

isee 发表于 2022-8-8 16:01 ((x)) 是约定的，还是只是这里临时定义的？

Unicode Character “⸨” (U+2E28)

Name:			Left Double Parenthesis
Unicode Version:	5.1 (April 2008)
Block:			Supplemental Punctuation, U+2E00 - U+2E7F
Plane:			Basic Multilingual Plane, U+0000 - U+FFFF
Script:			Code for undetermined script (Zyyy)
Category:			Open Punctuation (Ps)
Bidirectional Class:	Other Neutral (ON)
Combining Class:	Not Reordered (0)
Character is Mirrored:	Yes , ⸩ (U+2E29)
HTML Entity:		⸨	⸨
UTF-8 Encoding:	0xE2 0xB8 0xA8
UTF-16 Encoding:	0x2E28
UTF-32 Encoding:	0x00002E28