The three-fold integral J(t)=π−3∫0π∫0π∫0πlog⁡(t−cos⁡x−cos⁡y−cos⁡z+cos⁡xcos⁡ycos⁡z) dx dy dzJ(t)=\pi^{-3}\int_0^\pi\int_0^\pi\int_0^\pi\log(t-\cos x-\cos y-\cos z+\cos x\cos y\cos z)\,dx\,dy\,dz arises in the Bazhanov–Baxter solution of the three-dimensional sl(n)sl(n)-chiral Potts model: the model's thermodynamic free energy contains the term J(2)J(2), evaluated by those authors as 8πG−3log⁡2\frac{8}{\pi}G-3\log 2 with GG Catalan's constant. Guttmann and Rogers (2013) expressed J(t)J(t) through a combination of three interlocked 5F4_5F_4 hypergeometric functions, but proved the identity only for t≥2.0802…t\ge 2.0802\ldots on the real axis; beyond that, one hypergeometric argument crosses a branch cut. They asserted without proof a modular/lattice-sum expansion, stated to hold for v≥12v\ge\tfrac12 and to reproduce J(2)J(2), stated an unproved evaluation of J(5/2)J(5/2) in terms of the LL-series of the eta-product η3(2τ)η3(6τ)\eta^3(2\tau)\eta^3(6\tau), and announced further identities for J(14)J(14) and J(322)J(322) — all deferred to a companion paper that never appeared. The problem is to complete this program: prove the modular expansion, including convergence and its exact validity range; and establish the asserted special-value evaluations in terms of LL-series of modular forms (eta-products).

Background

Exactly solvable lattice models in two dimensions are organized by the Yang–Baxter equation, an algebraic consistency condition guaranteeing that transfer matrices commute and the model can be solved exactly. In three dimensions the analogous role is played by the tetrahedron equation, introduced by A. B. Zamolodchikov in 1980–81; R. J. Baxter solved Zamolodchikov's model in 1983 (his solution was published in 1986) using only commutativity, symmetry, and factorization properties of the transfer matrix. A decade later V. V. Bazhanov and R. J. Baxter studied the sl(n)sl(n)-chiral Potts model, a multi-state generalization of Zamolodchikov's model solved through its star-triangle relation (a form of the Yang–Baxter equation), and showed that in the thermodynamic limit the free energy contains the term J(2)J(2), where

J(t)=1π3∫0π∫0π∫0πlog⁡(t−cos⁡x−cos⁡y−cos⁡z+cos⁡xcos⁡ycos⁡z) dx dy dz.J(t)=\frac{1}{\pi^3}\int_0^\pi\int_0^\pi\int_0^\pi \log\left(t-\cos x-\cos y-\cos z+\cos x\cos y\cos z\right)\,dx\,dy\,dz.

For real t≥2t\ge 2 the argument of the logarithm is non-negative, so J(t)J(t) is real; for other tt it is defined by analytic continuation. Baxter and Bazhanov evaluated J(2)=8πL−4(2)−3log⁡2J(2)=\frac{8}{\pi}L_{-4}(2)-3\log 2, where Lk(s)=∑n≥1(kn)n−sL_{k}(s)=\sum_{n\ge1}\left(\frac{k}{n}\right)n^{-s} is a Dirichlet LL-series with (kn)\left(\frac{k}{n}\right) the Jacobi symbol (a quadratic Dirichlet character), and L−4(2)=∑n≥0(−1)n(2n+1)2L_{-4}(2)=\sum_{n\ge0}\frac{(-1)^n}{(2n+1)^2} is Catalan's constant GG.

The route to general tt runs through the related Green function G(t)G(t), the same integral with the logarithm replaced by the reciprocal of its argument. Joyce, Delves, and Zucker proved that for tt in the complex plane cut along [−2,2][-2,2],

G(t)=1t(1−4t2)−1/4[2F1(18,581;4t2)]2,G(t)=\frac{1}{t}\left(1-\frac{4}{t^2}\right)^{-1/4}\left[{}_2F_1\left(\begin{matrix}\tfrac18,\tfrac58\\ 1\end{matrix};\frac{4}{t^2}\right)\right]^2,

where 2F1{}_2F_1 is the Gauss hypergeometric function. Since J′(t)=G(t)J'(t)=G(t), Guttmann and Rogers integrated this formula. Here pFq(a1,…,apb1,…,bq;z)=∑n≥0(a1)n⋯(ap)n(b1)n⋯(bq)nznn!{}_pF_q\left(\begin{matrix}a_1,\ldots,a_p\\ b_1,\ldots,b_q\end{matrix};z\right)=\sum_{n\ge0}\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n}\frac{z^n}{n!} is the generalized hypergeometric function ((a)n(a)_n is the rising factorial); the defining series converges for ∣z∣<1|z|<1 and the function continues analytically with a branch cut on [1,∞)[1,\infty). Their Theorem 1 states that under the change of variable t=4α(1−α)+14α(1−α)t=\sqrt{4\alpha(1-\alpha)}+\frac{1}{\sqrt{4\alpha(1-\alpha)}},

J(t)=−12log⁡(4α(1−α)19(1+α)12)−114α(1−α) 5F4(32,32,32,1,12,2,2,2;4α(1−α))−7α4(1−α)25F4(32,32,32,1,12,2,2,2;−4α(1−α)2)+9α(1−α)24(1+α)45F4(54,32,74,1,12,2,2,2;16α(1−α)2(1+α)4).J(t)=-\tfrac12\log\left(4\alpha(1-\alpha)^{19}(1+\alpha)^{12}\right)-\tfrac{11}{4}\alpha(1-\alpha)\,{}_5F_4\left(\begin{matrix}\tfrac32,\tfrac32,\tfrac32,1,1\\ 2,2,2,2\end{matrix};4\alpha(1-\alpha)\right)-\tfrac{7\alpha}{4(1-\alpha)^2}{}_{5}F_{4}\left(\begin{matrix}\tfrac32,\tfrac32,\tfrac32,1,1\\ 2,2,2,2\end{matrix};-\tfrac{4\alpha}{(1-\alpha)^2}\right)+\tfrac{9\alpha(1-\alpha)^2}{4(1+\alpha)^4}{}_{5}F_{4}\left(\begin{matrix}\tfrac54,\tfrac32,\tfrac74,1,1\\ 2,2,2,2\end{matrix};\tfrac{16\alpha(1-\alpha)^2}{(1+\alpha)^4}\right).

The identity is proved only for ∣α∣|\alpha| sufficiently small; on the real axis this covers 0<α≤(2−1)2≈0.17150<\alpha\le(\sqrt2-1)^2\approx0.1715, i.e. t≥2.0802…t\ge 2.0802\ldots (the parametrization maps this interval of α\alpha onto that ray of tt). Beyond it the identity fails in its stated form because the argument 16α(1−α)2(1+α)4\frac{16\alpha(1-\alpha)^2}{(1+\alpha)^4} of the third 5F4_5F_4 crosses the cut [1,∞)[1,\infty). The paper notes that one may continue the identity along rays from α=0\alpha=0 "ending at a point where one of the functions ceases to be analytic", but the maximal continuation domain and the monodromy — the permutation of branches of the multi-valued right-hand side under continuation around its branch points, and how those branches match the branches of J(t)J(t) — are not described. Notably the physical value t=2t=2 lies just below the theorem's range.

To reach J(2)J(2) the authors state, without proof, a modular expansion: with q=e−2πvq=e^{-2\pi v} and

η(q)=q1/24∏n≥1(1−qn),u(q)=(24 η(q)η(q4)η2(q2))12+(24 η(q)η(q4)η2(q2))−12,\eta(q)=q^{1/24}\prod_{n\ge1}(1-q^n),\qquad u(q)=\left(\sqrt[4]{2}\,\frac{\eta(q)\eta(q^4)}{\eta^2(q^2)}\right)^{12}+\left(\sqrt[4]{2}\,\frac{\eta(q)\eta(q^4)}{\eta^2(q^2)}\right)^{-12},

where η\eta is the Dedekind eta function (the weight-one-half modular form q1/24∏(1−qn)q^{1/24}\prod(1-q^n)),

J(u(e−2πv))=−3log⁡2+15vπ3∑(n,k)≠(0,0)3n2−(2v)2k2(n2+(2v)2k2)3+48vπ3∑n,k3(2n+1)2−(2v)2(2k+1)2((2n+1)2+(2v)2(2k+1)2)3,J\left(u\left(e^{-2\pi v}\right)\right)=-3\log 2+\frac{15v}{\pi^3}\sum_{(n,k)\neq(0,0)}\frac{3n^2-(2v)^2k^2}{\left(n^2+(2v)^2k^2\right)^3}+\frac{48v}{\pi^3}\sum_{n,k}\frac{3(2n+1)^2-(2v)^2(2k+1)^2}{\left((2n+1)^2+(2v)^2(2k+1)^2\right)^3},

the sums running over integer pairs. The paper asserts that this holds for v≥12v\ge\tfrac12 and that the left-hand side equals J(2)J(2) when v=12v=\tfrac12, but defers the derivation, the convergence proof, and the exact validity range to a companion paper — cited as M. D. Rogers, "Evaluations of some three-variable Mahler measures", in progress — which never appeared.

The integral also carries an arithmetic meaning. The logarithmic Mahler measure of a Laurent polynomial PP is m(P)=∫01⋯∫01log⁡∣P(e2πit1,…,e2πitn)∣dt1⋯dtn\mathrm{m}(P)=\int_0^1\cdots\int_0^1\log\left|P\left(e^{2\pi it_1},\ldots,e^{2\pi it_n}\right)\right|dt_1\cdots dt_n, and Theorem 1 implies an identity expressing the Mahler measure of the family 84α(1−α)+84α(1−α)−4(x+x−1+y+y−1+z+z−1)+(x+x−1)(y+y−1)(z+z−1)8\sqrt{4\alpha(1-\alpha)}+\frac{8}{\sqrt{4\alpha(1-\alpha)}}-4(x+x^{-1}+y+y^{-1}+z+z^{-1})+(x+x^{-1})(y+y^{-1})(z+z^{-1}) as the combination 11 m(…)−7 m(…)−6 m(…)11\,\mathrm{m}(\ldots)-7\,\mathrm{m}(\ldots)-6\,\mathrm{m}(\ldots) of measures of the families (x+x−1)(y+y−1)(z+z−1)+c(x+x^{-1})(y+y^{-1})(z+z^{-1})+c and x4+y4+z4+1+c xyzx^4+y^4+z^4+1+c\,xyz. Since Boyd's experimental work, many three-variable Mahler measures are expected — and in special cases proved — to equal LL-values of modular forms: for a modular form f(τ)=∑n≥1ane2πinτf(\tau)=\sum_{n\ge1}a_n e^{2\pi i n\tau} (for example an eta-product, a product of powers of η\eta), its LL-series is L(f,s)=∑n≥1ann−sL(f,s)=\sum_{n\ge1}a_n n^{-s}. In this spirit the paper states, again without proof, J(52)=243π3L(η3(2τ)η3(6τ),3)+1534πL−3(2)−3log⁡2J\left(\tfrac52\right)=\frac{24\sqrt3}{\pi^3}L\left(\eta^3(2\tau)\eta^3(6\tau),3\right)+\frac{15\sqrt3}{4\pi}L_{-3}(2)-3\log2 and announces "more complicated identities" for J(14)J(14), J(322)J(322), and J(t)J(t) for various irrational algebraic tt, all deferred to the unpublished companion. The two 5F4_5F_4 functions appearing in Theorem 1 are the same as those in Guttmann and Rogers' evaluation of the spanning-tree constant of the simple-cubic lattice (the exponential growth rate of the number of spanning trees of large cubic-lattice regions).

Problem Statement

Complete the analytic and arithmetic theory of the integral J(t)=1π3∫0π∫0π∫0πlog⁡(t−cos⁡x−cos⁡y−cos⁡z+cos⁡xcos⁡ycos⁡z) dx dy dzJ(t)=\frac{1}{\pi^3}\int_0^\pi\int_0^\pi\int_0^\pi\log(t-\cos x-\cos y-\cos z+\cos x\cos y\cos z)\,dx\,dy\,dz that Guttmann and Rogers established only in fragmentary form. The two strands below concern the same integral and the same identity and constitute the program that paper deferred to its never-published companion; the common objective is a rigorous proof of the asserted modular expansion of J(t)J(t) and of the arithmetic content of its special values.

  1. Modular/lattice-sum expansion. Prove the expansion of J(u(e−2πv))J\left(u\left(e^{-2\pi v}\right)\right) as the two lattice sums displayed in the Background: supply the full derivation, prove convergence of the double sums, and determine exactly the range of vv for which the identity holds (the source asserts without proof that it holds for v≥12v\ge\tfrac12 and that v=12v=\tfrac12 recovers J(2)=8πL−4(2)−3log⁡2J(2)=\frac{8}{\pi}L_{-4}(2)-3\log 2).

  2. Special values and their arithmetic. Prove the asserted evaluation J(52)=243π3L(η3(2τ)η3(6τ),3)+1534πL−3(2)−3log⁡2J\left(\tfrac52\right)=\frac{24\sqrt3}{\pi^3}L\left(\eta^3(2\tau)\eta^3(6\tau),3\right)+\frac{15\sqrt3}{4\pi}L_{-3}(2)-3\log2; and establish the announced but never-published evaluations of J(14)J(14) and J(322)J(322).

References

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