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The Ising-class integral C5
Find a closed form for the Ising-class integral C5 = (4/15)∫ t K0(t)^5 dt ≈ 0.6657598002 of Bailey, Borwein and Crandall.
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| 1 | +# The Ising-class integral C5 | |
| 2 | + | |
| 3 | +In 2006 Bailey, Borwein and Crandall introduced the integrals $C_n$ as simpler relatives of the integrals that make up the magnetic susceptibility of the two-dimensional Ising model. $C_1$ to $C_4$ reduce to familiar constants such as $\zeta(3)$. The next one, $C_5 \approx 0.6657598$, has resisted identification ever since. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +For $n \ge 1$, let | |
| 8 | + | |
| 9 | +$$ | |
| 10 | +C_n = \frac{4}{n!} \int_0^\infty \!\!\cdots\! \int_0^\infty \frac{1}{\bigl(\sum_{j=1}^{n} (u_j + 1/u_j)\bigr)^2} \, \frac{du_1}{u_1} \cdots \frac{du_n}{u_n} = \frac{2^n}{n!} \int_0^\infty t \, K_0(t)^n \, dt, | |
| 11 | +$$ | |
| 12 | + | |
| 13 | +where $K_0$ is the modified Bessel function of the second kind. The second form is proved in [1]. Find a closed-form expression for | |
| 14 | + | |
| 15 | +$$ | |
| 16 | +C_5 = \frac{4}{15} \int_0^\infty t \, K_0(t)^5 \, dt = 0.66575980019993742831573380830706659819\ldots | |
| 17 | +$$ | |
| 18 | + | |
| 19 | +using standard constants and special functions. | |
| 20 | + | |
| 21 | +## Background | |
| 22 | + | |
| 23 | +Bailey, Borwein and Crandall studied three families, $C_n$, $D_n$ and $E_n$ [1]. The $D_n$ are susceptibility integrals: they enter the amplitudes of the magnetic susceptibility of the 2D Ising model near its critical temperature. $C_n$ is the same integral without the factor $\prod_{i<j} \bigl(\frac{u_i - u_j}{u_i + u_j}\bigr)^2$. Closed forms for $C_n$ were sought as a guide to the constants that appear in $D_n$. The $C_n$ also turned out to occur in bounds on certain amplitude transforms in Ising theory [1]. In Bessel form, $C_5$ belongs to the moments $\int_0^\infty t^k I_0(t)^a K_0(t)^b \, dt$ that arise as Feynman integrals in two-dimensional quantum field theory. With five Bessel functions, these moments are tied to a modular form of weight 3 and level 15 [3, 4]. | |
| 24 | + | |
| 25 | +## What is known | |
| 26 | + | |
| 27 | +- Proven values: $C_1 = 2$, $C_2 = 1$, $C_3 = L_{-3}(2) = \sum_{k \ge 0} \bigl( \frac{1}{(3k+1)^2} - \frac{1}{(3k+2)^2} \bigr)$, $C_4 = \frac{7}{12}\zeta(3)$, and $\lim_{n\to\infty} C_n = 2e^{-2\gamma}$ [1]. | |
| 28 | +- Bailey, Borwein and Crandall computed $C_5$ to 500 digits. With PSLQ they ruled out any integer relation of Euclidean norm below $4 \cdot 10^{12}$ between $C_5$ and more than 30 candidate constants, among them $\pi$, $\log 2$, $\zeta(3)$, $L_{-3}(2)$, Catalan's constant and $\mathrm{Li}_4(1/2)$. They asked whether a two-dimensional integral representation of $C_5$ can be evaluated symbolically [1]. | |
| 29 | +- Mellin–Barnes methods give closed forms for many Ising-class and box integrals. In 2024, however, the case $C_{5,k}$, which includes $C_5$, was still "not yet fully resolved" [2]. | |
| 30 | +- Broadhurst (2016) conjectured an evaluation of one linear combination of $\int_0^\infty t K_0^5 \, dt$ and $\int_0^\infty t^3 K_0^5 \, dt$, in terms of $\zeta(2)$ and L-values of that modular form. It does not determine $C_5$ by itself [3]. | |
| 31 | + | |
| 32 | +## What would count as progress | |
| 33 | + | |
| 34 | +- A closed-form expression that agrees with $C_5$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants. The reference value has 200 digits, and $C_5$ is easy to compute to hundreds of digits from the Bessel integral, so a candidate can be tested far beyond 20 digits. | |
| 35 | +- A proof of such an evaluation, or a proven formula that expresses $C_5$ through L-values or periods of the associated modular form. | |
| 36 | + | |
| 37 | +## References | |
| 38 | + | |
| 39 | +1. D. H. Bailey, J. M. Borwein and R. E. Crandall, "Integrals of the Ising class", *J. Phys. A* 39 (2006), 12271–12302. [doi:10.1088/0305-4470/39/40/001](https://doi.org/10.1088/0305-4470/39/40/001) | |
| 40 | +2. B. Ananthanarayan, T. Pathak and K. Sharma, "Closed form expressions for certain improper integrals of mathematical physics", *Eur. Phys. J. Spec. Top.* 233 (2024), 2057–2074. [doi:10.1140/epjs/s11734-024-01103-8](https://doi.org/10.1140/epjs/s11734-024-01103-8), [arXiv:2301.13436](https://arxiv.org/abs/2301.13436) | |
| 41 | +3. D. Broadhurst, "Feynman integrals, L-series and Kloosterman moments", *Commun. Number Theory Phys.* 10 (2016), 527–569. [doi:10.4310/CNTP.2016.v10.n3.a3](https://doi.org/10.4310/CNTP.2016.v10.n3.a3), [arXiv:1604.03057](https://arxiv.org/abs/1604.03057) | |
| 42 | +4. K. Acres and D. Broadhurst, "Empirical determinations of Feynman integrals using integer relation algorithms", 2021. [arXiv:2103.06345](https://arxiv.org/abs/2103.06345) | |
| 43 | + | |
| 44 | +## Source | |
| 45 | + | |
| 46 | +Imported from [HorizonMath c5_ising_susceptibility](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L572-L582), curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. | |