"3-manifolds and their invariants"의 두 판 사이의 차이

2010년 8월 1일 (일) 04:52 판

Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
Bloch-Wigner dilogarithm is involved

Prove
\(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))\)
a log tangent integral

[[4909919|]]

Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links
- J.M. Borwein, D.J. Broadhurst, 1998
Three-manifolds and the Temperley-Lieb algebra
- W. B. R. Lickorish, 1991
Hyperbolic manifolds and special values of Dedekind zeta-functions
- Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월

@@ 19번째 줄: / 19번째 줄: @@
-<h5 style="line-height: 2em; margin: 0px;">an open problem</h5>
+<h5 style="line-height: 2em; margin: 0px;">a problem</h5>
 *  Prove<br><math>\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))</math><br>
-* [[a log tangent integral|problems in integrals]]<br>
+* [[a log tangent integral]]<br>