戴德金和戴德金和(Dedekind sum)是德国数学家理查德·戴德金在跟戴德金η函数有关的工作中提出的。 定义这个函数,首先要定义 ( ( x ) ) {\displaystyle ((x))} :若 x {\displaystyle x} 是整数, ( ( x ) ) = 0 {\displaystyle ((x))=0} ,否则为 x − [ x ] − 0.5 {\displaystyle x-[x]-0.5} ,其中 [ x ] {\displaystyle [x]} 是最大而又不大于 x {\displaystyle x} 的整数。 对于非零整数 h , k {\displaystyle h,k} ,戴德金和 s ( h , k ) {\displaystyle s(h,k)} 定义为 s ( h , k ) = ∑ μ = 0 k − 1 ( ( μ k ) ) ( ( h μ k ) ) {\displaystyle s(h,k)=\sum _{\mu =0}^{k-1}(({\frac {\mu }{k}}))(({\frac {h\mu }{k}}))} 若 h , k {\displaystyle h,k} 互质且均大于0,有 s ( h , k ) = 1 4 k ∑ μ = 1 k − 1 cot ( π h μ k ) cot ( π μ k ) {\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 有公因数时: s ( c h , c k ) = s ( h , k ) {\displaystyle s(ch,ck)=s(h,k)} Petersson-Knopp恒等式: ∑ d | n ∑ m = 0 d − 1 s ( n d h + m k , k d ) = σ ( n ) s ( h , k ) {\displaystyle \sum _{d|n}\sum _{m=0}^{d-1}s\left({\frac {n}{d}}h+mk,kd\right)=\sigma (n)s(h,k)} , σ ( n ) {\displaystyle \sigma (n)} 为因数函数,是 n {\displaystyle n} 的正因数之和。其中一个较易证明的特例为当 p {\displaystyle p} 为质数, ( p + 1 ) s ( h , k ) = s ( p h , k ) + ∑ m = 0 p − 1 s ( h + m k , p k ) {\displaystyle (p+1)s(h,k)=s(ph,k)+\sum _{m=0}^{p-1}s(h+mk,pk)} 周期性: s ( n k + h , k ) = s ( h , k ) {\displaystyle s(nk+h,k)=s(h,k)} 若 p q ≡ 1 ( mod k ) {\displaystyle pq\equiv 1{\pmod {k}}} , s ( p , k ) = s ( q , k ) {\displaystyle s(p,k)=s(q,k)} 。 s ( 1 , k ) = ( k − 1 ) ( k − 2 ) 12 k {\displaystyle s(1,k)={\frac {(k-1)(k-2)}{12k}}} 若 k {\displaystyle k} 为奇数, s ( 2 , k ) = ( k − 1 ) ( k − 5 ) 24 k {\displaystyle s(2,k)={\frac {(k-1)(k-5)}{24k}}} 对于 k ≡ 1 ( mod h ) {\displaystyle k\equiv 1{\pmod {h}}} , 12 h k s ( h , k ) = ( k − 1 ) ( k − ( h 2 + 1 ) ) {\displaystyle 12hks(h,k)=(k-1)(k-(h^{2}+1))} 对于 k ≡ 2 ( mod h ) {\displaystyle k\equiv 2{\pmod {h}}} , 12 h k s ( h , k ) = ( k − 2 ) ( k − ( h 2 + 1 ) / 2 ) {\displaystyle 12hks(h,k)=(k-2)(k-(h^{2}+1)/2)} 对于 k ≡ − 1 ( mod h ) {\displaystyle k\equiv -1{\pmod {h}}} , 12 h k s ( h , k ) = k 2 + ( h 2 − 6 h + 2 ) k + ( h 2 + 1 ) {\displaystyle 12hks(h,k)=k^{2}+(h^{2}-6h+2)k+(h^{2}+1)} 互反和: s ( h , k ) + s ( k , h ) = − 1 4 + 1 12 ( h k + 1 h k + k h ) {\displaystyle s(h,k)+s(k,h)=-{\frac {1}{4}}+{\frac {1}{12}}\left({\frac {h}{k}}+{\frac {1}{hk}}+{\frac {k}{h}}\right)} 参考 https://web.archive.org/web/20070929120859/http://gifted.hkedcity.net/Gifted/Download/notes/0607math2phase/advanced/06-11-4-11-18_dedekind%20sums.pdf http://mathworld.wolfram.com/DedekindSum.html (页面存档备份,存于互联网档案馆) http://arxiv.org/abs/math/0112077 (页面存档备份,存于互联网档案馆)