The five-dimensional Watson integral

A simple random walk on the five-dimensional integer lattice returns to its starting point with probability about 0.135. That probability is fixed by the Watson integral $W_5 \approx 0.2312616$, which is, up to normalization, the lattice Green function of $\mathbb{Z}^5$ at the origin. Watson's integral has a closed form in three dimensions, and since 2019 one in four dimensions as well. In five dimensions none is known.

Problem

For $d \ge 3$ let

$$W_d = \frac{1}{\pi^d} \int_0^\pi \!\!\cdots\! \int_0^\pi \frac{dx_1 \cdots dx_d}{d - \cos x_1 - \cdots - \cos x_d} = \int_0^\infty e^{-dt} \, I_0(t)^d \, dt,$$

where $I_0$ is the modified Bessel function. The second form follows from $1/a = \int_0^\infty e^{-at}\,dt$ and $\frac{1}{\pi}\int_0^\pi e^{t\cos x}\,dx = I_0(t)$. Find a closed-form expression for

$$W_5 = 0.23126162496804623574142702438771339710\ldots$$

in terms of standard constants and special functions. The probability that the simple random walk on $\mathbb{Z}^d$ ever returns to the origin is $1 - 1/(d\,W_d)$ [3], which is about $0.13518$ for $d = 5$.

Background

The product $d\,W_d$ is the hypercubic lattice Green function $P(\vec 0; z)$ at $z = 1$. Such Green functions give return probabilities of random walks and effective resistances of resistor networks [3]. For $d = 3$, Watson evaluated the cubic-lattice integrals in 1939 [1], and Glasser and Zucker reduced the simple-cubic case to a product of Gamma values in 1977 [2]. Laporta conjectured that the four-dimensional integral is, up to a constant, a four-loop sunrise Feynman diagram of two-dimensional quantum field theory. Zhou proved this, together with a reduction to special values of Euler's Gamma function and generalized hypergeometric series [4]. Five dimensions is the next case.

What is known

• $d = 3$: Watson (1939) evaluated the integral with elliptic integrals [1], and Glasser and Zucker (1977) gave a closed form in Gamma values [2, 3].
• $d = 4$: Laporta's conjectured formulae were proved by Zhou (2019). They link $W_4$ to the four-loop sunrise integral and reduce it to Gamma values and generalized hypergeometric series [4].
• $d = 5$: Guttmann (2010) treated lattice Green functions in all dimensions. The five-dimensional hypercubic one satisfies a linear differential equation of order 5 [3]. No closed form for $W_5$ is known. HorizonMath's reference value has 110 digits.

What would count as progress

• A closed-form expression that agrees with $W_5$ 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 Bessel form lets anyone compute $W_5$ to many more digits, so candidates can be tested far beyond 20.
• A proof of such an evaluation, or an exact reduction of $W_5$ to Bessel moments or periods that are already understood.

References

1. G. N. Watson, "Three triple integrals", Q. J. Math. 10 (1939), 266–276. doi:10.1093/qmath/os-10.1.266
2. M. L. Glasser and I. J. Zucker, "Extended Watson integrals for the cubic lattices", Proc. Natl. Acad. Sci. USA 74 (1977), 1800–1801. doi:10.1073/pnas.74.5.1800
3. A. J. Guttmann, "Lattice Green's functions in all dimensions", J. Phys. A 43 (2010), 305205. doi:10.1088/1751-8113/43/30/305205, arXiv:1004.1435
4. Y. Zhou, "On Laporta's 4-loop sunrise formulae", Ramanujan J. 50 (2019), 465–503. doi:10.1007/s11139-018-0090-z, arXiv:1801.02182

Source

Imported from HorizonMath w5_watson_integral, 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.