Broadhurst–Mellit vacuum determinants

—   total number of Bessel factors ($n\in\{5,6,8\}$ for the three constructions defined here)

$V_5=\frac{2\pi^3}{15\sqrt{15}}\mu_5$ and $V_6=\frac{\pi^4}{96}\mu_6$, where $\mu_5$ and $\mu_6$ are the logarithmic Mahler measures $\mu_n=m(1+x_1+\cdots+x_{n-1})$ [1], Theorem 1.3.

For $n=5,6$, $V_n=n\mu_nD_n$, where $D_n$ is the full-period determinant and $\mu_n$ is the logarithmic Mahler measure in (1). Comparing the product formulas for $D_n$ with the Mahler-measure formulas gives the factors $5$ and $6$ exactly [1], equations (1.4), (1.6), (1.9) and (1.10).

The unit-scale Bessel moment is $B_n(a,r)=\int_0^\infty I_0(t)^aK_0(t)^{n-a}t^r\,dt=\operatorname{IKM}(a,n-a;r)$, where $I_0$ and $K_0$ are the standard order-zero modified Bessel functions on the positive real axis. Every factor has argument $t$. No loop factors, powers of $\pi$ or factorials are absorbed into the moments.

The determinant is signed, with the vacuum row $a=0$ first and the row $a=2$ second; the $r=1$ column comes first. For $n=8$, the second column integrates $t^3-2t^5$. It is not the monomial-column determinant obtained by continuing the five- and six-factor construction.

All moments converge without regularization. At zero the integrands are $O(t^r(1+|\log t|)^{n-a})$ with $r\geq1$; at infinity they are $O(t^{r-n/2}e^{-(n-2a)t})$ with $n>2a$. The polynomial combination at $n=8$ retains this exponential decay.

Zhou denotes $V_5$ and $V_6$ by $\det\check{\mathbf M}_2$ and $\det\check{\mathbf N}_2$ in Theorem 1.3 of [1]; its proof is available in [3]. Broadhurst defines $V_8=\det\mathcal M_2$ in Section 7.8, Conjecture 6 of [2], which also states its modular $L$-value evaluation. The notation $V_n$ here means a determinant, whereas Zhou also uses $V_n$ for the scalar vacuum integral $B_n(0,1)$.

These are three specified vacuum-containing matrices, not a construction defined at every factor count. They are not matrix entries used to compute the full-period determinants: both families are assembled from individual Bessel moments. The full-period family uses consecutive positive-$a$ rows and monomial columns and is defined for every $n\geq3$. The vacuum constructions cited here do not specify a matrix for $n=7$ or every $n>8$.

Python (mpmath)

from functools import lru_cache
from mpmath import mp
mp.dps = 50

@lru_cache(maxsize=None)
def bessel(t):
    return mp.besseli(0, t), mp.besselk(0, t)

def B(n, a, r):
    def integrand(t):
        if not t:
            return mp.mpf(0)
        i, k = bessel(t)
        return i**a * k**(n-a) * t**r
    return mp.quad(integrand, [0, 1, 4, 16, 64, mp.inf])

def V(n):
    if n not in (5, 6, 8):
        raise ValueError('The defined factor counts are 5, 6, 8')
    def C(a):
        return B(n, a, 3) - (2*B(n, a, 5) if n == 8 else 0)
    return B(n, 0, 1)*C(2) - B(n, 2, 1)*C(0)

for n in (5, 6, 8):
    print(n, mp.nstr(V(n), 30))

Yajun Zhou, Wrońskian factorizations and Broadhurst–Mellit determinant formulae (2018), Theorem 1.3, equations (1.9) and (1.10), and Section 5. (arXiv)

David Broadhurst, Feynman integrals, L-series and Kloosterman moments (2016), Sections 7.2, 7.3 and 7.8; Conjecture 6 in Section 7.8 specifies the eight-factor matrix. (arXiv)

Full text of Zhou’s determinant theorem

Bessel moments —   the individual unit-scale integrals used in the determinant matrices; the five- and six-factor entries occur there with parameters $(a,n-a,r)$

Broadhurst–Mellit determinants of Bessel moments —   the full-period matrices with consecutive positive first-kind exponents and monomial columns

Mahler measures of sums of monomials —   the five- and six-term logarithmic measures related to the vacuum determinants by Zhou’s theorem

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds the three vacuum constructions specified in the cited sources, at $n=5,6,8$. The five- and six-factor determinants use the monomial columns in Zhou’s Theorem 1.3, and the eight-factor determinant uses Broadhurst’s specified polynomial column. This is a source-defined selection, not a numerical-precision or computation-time cutoff; adding other factor counts would require specifying and motivating additional matrices, rather than simply extending the range of the full-period family)

How they were obtained:

The defining moments were computed by tanh-sinh quadrature at 150 and 200 working decimal digits, retaining 100 significant digits. Arb evaluates the Bessel functions with their relative radii checked before their midpoints enter quadrature. The largest relative discrepancy between the determinant runs is below $3.8\times10^{-151}$; cancellation in the final two-product subtraction loses less than $0.245$ decimal digits. The integral is split at $1,4,16,64$ and truncated at $3(w+60)$, where $w$ is the working precision. The generator gives an analytic exponential upper-tail estimate. The quadrature itself is not a certified enclosure, so the rigour is agreement-checked.

more

Before the table moments, the nonzero controls $\int_0^\infty tK_0(t)\,dt=1$ and $\int_0^\infty tK_0(t)^2\,dt=1/2$ were tested. The five- and six-factor full-period determinants were also compared with Zhou’s proved products. Independent calculations from eta-product Fourier coefficients and split Mellin transforms at 130 and 180 working digits agree with all three vacuum determinants to more than 180 relative digits. Those computations share no Bessel evaluation or moment quadrature code with the generator. Their Euler-product coefficients, Mellin normalization and imaginary-axis modular transformations have separate controls. The modular determinant relations in Broadhurst’s paper are used as checks, not as certificates for the stored digits.

The attached argument-selection.md explains the finite catalogue. The generator, standalone checks and numerical evidence record the methods, dependency versions, precision comparisons and archived skill manifest.