후르비츠 수 (Hurwitz number)

$K=Q(\sqrt{-1})$의 예

• 다음이 성립한다

$$ \sum_{ (m,n)\in \mathbb{Z}^2\backslash\{(0,0)\}} \frac{1}{(mi+n)^{4s}}=G_{4s}\omega^{4s},\,s=1,2,\cdots $$ 여기서 $G_{s}$는 다음과 같은 상수 $$ \begin{array}{c|cccccccc} s & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 \\ \hline G_s & \frac{1}{15} & \frac{1}{525} & \frac{2}{53625} & \frac{1}{1243125} & \frac{2}{118096875} & \frac{2}{5575415625} & \frac{4}{527240390625} & \frac{223}{1389278429296875} \\ \end{array} $$

관련논문

• Bannai, Kenichi, and Shinichi Kobayashi. “Integral Structures on $p$-Adic Fourier Theory.” arXiv:0804.4338 [math], April 28, 2008. http://arxiv.org/abs/0804.4338.
• Katz, Nicholas M. “The Congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz Numbers.” Mathematische Annalen 216, no. 1 (July 1, 1975): 1–4. doi:10.1007/BF02547966.