The Sondow–Zlobin conjecture on triangle integrals: prove that $I_n$ is never a rational multiple of $\zeta(n+2)$ for any integer $n>2$

The Sondow–Zlobin integrals $I_n=\iint_T (-\ln xy)^n/(xy)\,dx\,dy$ over the triangle $T=\{(x,y)\in[0,1]^2 : x+y\ge 1\}$ generalize an integral evaluated by Euler ( $I_0=\zeta(2)$ ). The first values are rational multiples of a single zeta value, $I_0=\zeta(2)$ , $I_1=2\zeta(3)$ , $I_2=\tfrac{9}{2}\zeta(4)$ , while for $n\ge 3$ one gets polynomials in zeta values such as $I_3=36\zeta(5)-12\zeta(2)\zeta(3)$ and $I_4=\tfrac{237}{2}\zeta(6)-48\zeta(3)^2$ . Sondow and Zlobin conjectured that this collapse ends at $n=2$ : $I_n$ is never a rational multiple of $\zeta(n+2)$ for $n>2$ . They proved this only conditionally, assuming algebraic independence of $\pi,\zeta(3),\zeta(5),\dots$ over $\mathbb{Q}$ . The conjecture remains open for every individual $n\ge 3$ .

Background

A classical theme of number theory is the arithmetic nature of special values of the Riemann zeta function $\zeta(s)=\sum_{k\ge 1}k^{-s}$ . Euler proved that $\zeta(2)=\pi^2/6$ and that every even value $\zeta(2m)$ is a rational multiple of $\pi^{2m}$ ; far less is known about odd values. It is classical that $\zeta(3)$ is irrational (Apéry's theorem; for a published account see Beukers, who also recast the proofs through double integrals over the unit square). Rivoal proved that infinitely many of $\zeta(3),\zeta(5),\zeta(7),\dots$ are irrational, and Zudilin proved that at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational; but no single odd value besides $\zeta(3)$ is known to be irrational, and nothing is known about irrationality or algebraic independence for ratios of products such as $\pi^2\zeta(3)/\zeta(5)$ (numbers are algebraically independent over $\mathbb{Q}$ when they satisfy no nontrivial polynomial equation with rational coefficients).

A rich supply of related constants comes from multiple zeta values (MZVs), the nested sums $\zeta(s_1,\dots,s_l)=\sum_{n_1>\cdots>n_l\ge 1}\frac{1}{n_1^{s_1}\cdots n_l^{s_l}},\qquad s_1\ge 2,$ which generalize $\zeta(s)$ ; their total weight is $s_1+\cdots+s_l$ , and they satisfy many linear identities, for example the duality $\zeta(k+2,\{1\}_l)=\zeta(l+2,\{1\}_k)$ , where $\{1\}_l$ denotes a string of $l$ ones.

Euler evaluated the integral of $1/(xy)$ over the triangle $T=\{(x,y)\in[0,1]^2 : x+y\ge 1\}$ with vertices $(1,0),(0,1),(1,1)$ , finding $\zeta(2)$ . Sondow and Zlobin systematically studied the family $I_n=\iint_T \frac{(-\ln xy)^n}{xy}\,dx\,dy,\qquad n=-1,0,1,2,\dots$ For $n\ge 0$ they proved $I_n=n!\sum_{k=0}^{n}\zeta(n-k+2,\{1\}_k)$ and showed that $I_n$ equals an explicit polynomial with rational coefficients in $\zeta(2),\zeta(3),\dots,\zeta(n+2)$ (based on Zlobin's earlier note, which gives for instance $I_3=36\zeta(5)-12\zeta(2)\zeta(3)$ and $I_4=\tfrac{237}{2}\zeta(6)-48\zeta(3)^2$ ). The first three values collapse to single zeta values, $I_0=\zeta(2)$ , $I_1=2\zeta(3)$ , $I_2=\tfrac{9}{2}\zeta(4)$ , each a rational multiple of $\zeta(n+2)$ . On the basis of these cases, the explicit formulas for larger $n$ , and numerical evidence, they conjectured that this collapse ends exactly at $n=2$ (their Conjecture 1), noting that the statement "has not been proved for a single value of $n$ ". They proved the conjecture for all $n>2$ only under the hypothesis that $\pi,\zeta(3),\zeta(5),\zeta(7),\dots$ are algebraically independent over $\mathbb{Q}$ (their Theorem 2) — a standard but far stronger conjecture in transcendence theory that they cite, not pose. The same paper also asks for an evaluation of the adjacent member $I_{-1}=\iint_T \frac{dx\,dy}{xy(-\ln xy)}=1.7330025\dots$ in terms of more familiar constants, relates it to Euler's constant $\gamma$ (the limit $\lim_{n\to\infty}(1+\tfrac12+\cdots+\tfrac1n-\ln n)$ ), and extends $I_n$ to integrals over higher-dimensional polytopes, deriving identities between multiple polylogarithm values at $1/2$ (values of the iterated sums $\mathrm{Li}_{s_1,\dots,s_l}(z)=\sum_{n_1>\cdots>n_l\ge 1} z^{n_1}/(n_1^{s_1}\cdots n_l^{s_l})$ , generalizing the polylogarithm $\mathrm{Li}_s(z)=\sum_{k\ge 1}z^k/k^s$ ) and MZVs. The conjecture at issue concerns the arithmetic, not the evaluation, of the $I_n$ .

Problem Statement

Let $T=\{(x,y)\in[0,1]^2 : x+y\ge 1\}$ be the triangle with vertices $(1,0)$ , $(0,1)$ , $(1,1)$ , and define $I_n=\iint_T \frac{(-\ln xy)^n}{xy}\,dx\,dy .$ Conjecture (Sondow–Zlobin, Conjecture 1): the integral $I_n$ is not a rational multiple of $\zeta(n+2)$ when $n>2$ ; equivalently, $I_n/\zeta(n+2)\notin\mathbb{Q}$ for every integer $n\ge 3$ .

Resolve this conjecture: either prove that $I_n$ is not a rational multiple of $\zeta(n+2)$ for all integers $n>2$ , or refute it by exhibiting an integer $n>2$ and a rational number $q$ , together with a rigorous proof of the identity $I_n=q\,\zeta(n+2)$ . The conjecture is currently unproved for even a single value of $n$ ; it is known to hold for all $n>2$ assuming algebraic independence over $\mathbb{Q}$ of $\pi,\zeta(3),\zeta(5),\dots$ (Sondow–Zlobin Theorem 2), and unconditional resolution is sought. Using the source paper's evaluations, the first two cases are equivalent to $\pi^2\zeta(3)/\zeta(5)\notin\mathbb{Q}$ (case $n=3$ , from $I_3=36\zeta(5)-12\zeta(2)\zeta(3)$ and $\zeta(2)=\pi^2/6$ ) and $\zeta(3)^2/\zeta(6)\notin\mathbb{Q}$ (case $n=4$ , from $I_4=\tfrac{237}{2}\zeta(6)-48\zeta(3)^2$ ).

References

Jonathan Sondow and Sergey Zlobin, "Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant", Mathematical Notes 84 (2008) 568–583 (Russian original: Matematicheskie Zametki 84 (2008) 609–626); DOI 10.1134/S0001434608090290; arXiv:0705.0732, https://arxiv.org/abs/0705.0732
Sergey Zlobin, "On a Certain Integral Over a Triangle" (2005), arXiv:math/0511239, https://arxiv.org/abs/math/0511239
Sergey Zlobin, "Special values of generalized polylogarithms", Journal of Mathematical Sciences 182 (2012) 484–504; DOI 10.1007/s10958-012-0752-0; arXiv:0712.1656, https://arxiv.org/abs/0712.1656
Jeffrey C. Lagarias, "Euler's constant: Euler's work and modern developments", Bulletin of the American Mathematical Society 50 (2013) 527–628; DOI 10.1090/S0273-0979-2013-01423-X; arXiv:1303.1856, https://arxiv.org/abs/1303.1856
Iaroslav V. Blagouchine, "Three Notes on Ser's and Hasse's Representations for the Zeta-functions" (2016), arXiv:1606.02044, https://arxiv.org/abs/1606.02044
F. Beukers, "A Note on the Irrationality of ζ(2) and ζ(3)", Bulletin of the London Mathematical Society 11 (1979) 268–272; DOI 10.1112/blms/11.3.268, https://doi.org/10.1112/blms/11.3.268
T. Rivoal, "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs", Comptes Rendus de l'Académie des Sciences – Series I – Mathematics 331 (2000) 267–270; DOI 10.1016/S0764-4442(00)01624-4, https://doi.org/10.1016/S0764-4442(00)01624-4
Wadim Zudilin, "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational", Russian Mathematical Surveys 56 (2001) 774–776; DOI 10.1070/RM2001V056N04ABEH000427, https://doi.org/10.1070/RM2001V056N04ABEH000427 (Russian original: Uspekhi Matematicheskikh Nauk 56 (2001) 149–150; DOI 10.4213/rm427)