The Ising-class integral C5

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.

Problem

For $n \ge 1$, let

$$C_n = \frac{4}{n!} \int_0^\infty \!\!\cdots\! \int_0^\infty \frac{1}{\bigl(\sum_{j=1}^{n} (u_j + 1/u_j)\bigr)^2} \, \frac{du_1}{u_1} \cdots \frac{du_n}{u_n} = \frac{2^n}{n!} \int_0^\infty t \, K_0(t)^n \, dt,$$

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

$$C_5 = \frac{4}{15} \int_0^\infty t \, K_0(t)^5 \, dt = 0.66575980019993742831573380830706659819\ldots$$

using standard constants and special functions.

Background

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].

What is known

• 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].
• 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].
• 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].
• 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].

What would count as progress

• 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.
• A proof of such an evaluation, or a proven formula that expresses $C_5$ through L-values or periods of the associated modular form.

References

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
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, arXiv:2301.13436
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, arXiv:1604.03057
4. K. Acres and D. Broadhurst, "Empirical determinations of Feynman integrals using integer relation algorithms", 2021. arXiv:2103.06345

Source

Imported from HorizonMath c5_ising_susceptibility, 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.