다이로그 항등식 (dilogarithm identities)

원분다항식 (단위근에 대한 원분다항식(cyclotomic polynomial) 과는 의미가 다르나 이미 용어가 굳어져 널리 사용됨)

$\prod_{r}(1-x^r)^{e_r}=x^{e_0}$

는 정수, 은 자연수
대응되는 dilogarithm 항등식

$\sum_{r}\frac{e_r}{r}L(x^r)=cL(1)$

여기서 c 는 유리수

오일러

$L(1)=\frac{\pi^2}{6}$

$2L(\frac{1}{2})=L(1)$
Lewin

$L(\frac{1}{2^6})-2L(\frac{1}{2^3})-6L(\frac{1}{2^2})+2L(1)=0$

$L(\frac{1}{3^2})-6L(\frac{1}{3})+2L(1)=0$
???

$2 L\left(\frac{1}{p}\right)+\sum _{j=2}^{p-1} L\left(\frac{1}{j^2}\right)=L(1)$

란덴

$5L(\frac{3-\sqrt{5}}{2})=2L(1)$

$5L(\frac{-1+\sqrt{5}}{2})=3L(1)$
콕세터(1935)

$\rho=\tfrac{1}{2}(\sqrt{5}-1)$ 는 황금비

$L(\rho^6)=4L(\rho^3)+3L(\rho^2)-6L(\rho)+\frac{7\pi^2}{30}$

$L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)+\frac{7\pi^2}{10}$

$L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{\pi^2}{5}$
Lewin

$L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{\pi^2}{30}$
Lewin

$x=\frac{\sqrt{13}-3}{2}$
Browkin

$x=\frac{\sqrt{13}-1}{6}$ , $z=\frac{\sqrt{13}+1}{6}$
르장드르 카이 함수

$L(\sqrt{2}-1)-4L((\sqrt{2}-1)^2)=\frac{\pi^2}{4}$

$L(\frac{\sqrt{5}-1}{2})-4L((\frac{\sqrt{5}-1}{2})^2)=\frac{\pi^2}{3}$

$L(\sqrt{5}-2)-4L((\sqrt{5}-2)^2)=\frac{\pi^2}{6}$
Loxton

$x=\frac{\sqrt{3}-1}{2}$

$12L(\beta)+3L(\beta^{2})-2L(\beta^{3})=\frac{5\pi^{2}}{6}$

$\sum_{i=1}^{k}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{3k}{k+2}\cdot\frac{\pi^2}{6}$

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