# 戴德金和

${\displaystyle h,k}$互質且均大於0，有${\displaystyle s(h,k)={\frac {1}{4k}}\sum _{\mu =1}^{k-1}\cot \left({\frac {\pi h\mu }{k}}\right)\cot \left({\frac {\pi \mu }{k}}\right)}$

## 公式

• 公因數時：${\displaystyle s(ch,ck)=s(h,k)}$
• Petersson-Knopp恆等式：${\displaystyle \sum _{d|n}\sum _{m=0}^{d-1}s\left({\frac {n}{d}}h+mk,kd\right)=\sigma (n)s(h,k)}$ ${\displaystyle \sigma (n)}$ 因數函數，是${\displaystyle n}$ 的正因數之和。其中一個較易證明的特例為當${\displaystyle p}$ 質數${\displaystyle (p+1)s(h,k)=s(ph,k)+\sum _{m=0}^{p-1}s(h+mk,pk)}$
• 周期性：${\displaystyle s(nk+h,k)=s(h,k)}$
• ${\displaystyle pq\equiv 1{\pmod {k}}}$ ${\displaystyle s(p,k)=s(q,k)}$
• ${\displaystyle s(1,k)={\frac {(k-1)(k-2)}{12k}}}$
• ${\displaystyle k}$ 奇數${\displaystyle s(2,k)={\frac {(k-1)(k-5)}{24k}}}$
• 對於${\displaystyle k\equiv 1{\pmod {h}}}$ ${\displaystyle 12hks(h,k)=(k-1)(k-(h^{2}+1))}$
• 對於${\displaystyle k\equiv 2{\pmod {h}}}$ ${\displaystyle 12hks(h,k)=(k-2)(k-(h^{2}+1)/2)}$
• 對於${\displaystyle k\equiv -1{\pmod {h}}}$ ${\displaystyle 12hks(h,k)=k^{2}+(h^{2}-6h+2)k+(h^{2}+1)}$
• 互反和：
${\displaystyle s(h,k)+s(k,h)=-{\frac {1}{4}}+{\frac {1}{12}}\left({\frac {h}{k}}+{\frac {1}{hk}}+{\frac {k}{h}}\right)}$