# Equivalence of two expressions for the Weeks manifold hyperbolic volume

An expression for the hyperbolic volume of the Weeks manifold is given on Wikipedia: $$\frac{3\cdot23^{3/2}\zeta_K(2)}{4\pi^4}=0.94270736277\dots\tag1$$ Here $$\zeta_K$$ is the Dedekind zeta function corresponding to the number field $$t^3-t+1$$. This can be easily evaluated in PARI/GP:

? 3*23^(3/2)*lfun(x^3-x+1,2)/(4*Pi^4)
%8 = 0.94270736277692772092129960309221164758


When I edited the corresponding OEIS entry A126774 in 2016 to include the 2009 proof that the Weeks manifold minimises hyperbolic volume among closed orientable hyperbolic $$3$$-manifolds, there was already a comment giving an alternative formula for the volume from a Herman Jamke: $$\operatorname{Im}(\operatorname{Li}_2(t_0)-\ln|t_0|\ln(1-t_0))=0.94270736277\dots\tag2$$ Here $$t_0$$ is the root with positive imaginary part of $$t^3-t+1$$. This is also easily evaluable in PARI/GP – and mpmath and Mathematica and other software since there is no Dedekind zeta function involved:

? th = polroots(x^3-x+1)[3];
? imag(dilog(th) + log(abs(th))*log(1-th))
%2 = 0.94270736277692772092129960309221164759


No reference whatsoever had been provided for the equivalence between these two expressions in the nine years between Jamke's comment and my OEIS edit, and there is still none. While I simply shrugged and moved on back then, I have gotten a bit into zeta and $$L$$-functions from needing to compute the Dirichlet/analytic density in Chebotarev's density theorem (cf. here, which uses the not-always-defined natural density).

How can $$(2)$$ be derived from $$(1)$$?

• Related: The article D.Zagier "The Dilogarithm Function" in "Frontiers in Number Theory, Physics, and Geometry." II pp. 3-65. However, one needs extra work to connect it to the OP since the zeta-function in Zagier's paper is associated with a quadratic extension of ${\mathbb Q}$ (because the hyperbolic orbifolds in Zagier's paper are the ones associated with Bianchi groups), while the zeta-function in OP is for a cubic extension of ${\mathbb Q}$. May 31, 2021 at 2:18