Mahler measures of $x+x^{-1}+y+y^{-1}+z+z^{-1}-k$

—   coefficient parameter ($k\geq0$)

$\displaystyle m(P_k)=\int_{-4}^4 \rho(s)J(k-s)\,ds$, where $\displaystyle \rho(s)=\frac{K(1-s^2/16)}{2\pi^2}$ and $\displaystyle J(a)=\begin{cases}0,&|a|\leq2,\\\operatorname{arcosh}(|a|/2),&|a|>2.\end{cases}$ Here $K(m)$ uses the parameter $m$, and $\rho$ is the probability density of $2\cos u+2\cos v$ for independent uniform torus angles $u,v$. Jensen's formula in $z$ gives this integral; the logarithmic singularity of $\rho$ at zero is integrable.

$\displaystyle m(P_k)=\log k-\sum_{n=1}^{\infty}\frac{c_n}{2n k^{2n}},\qquad c_n=\binom{2n}{n}\sum_{j=0}^n\binom nj^2\binom{2j}{j},\qquad k>6.$ The integer $c_n$ counts closed walks of length $2n$ from the origin in the simple-cubic lattice [3], Section 5.3.

$\displaystyle m(P_0)=m(1+x+y)=\frac{3\sqrt3}{4\pi}L(2,\chi_{-3})$ [1], Section 1. The three-step Mahler measure and $L(2,\chi_{-3})$ use the character taking $0,1,-1$ on residues $0,1,2$ modulo $3$.

$m(P_{10})=3m(P_2)+2m(P_0)$, obtained by eliminating the common weight-three modular $L$-value between the evaluations of $m(P_2)$ and $m(P_{10})$ in [1], Section 1.

$\displaystyle m(P_6)=h_{\mathrm{sc}},\qquad 6\left.\frac{d}{dk}m(P_k)\right|_{k=6^+}=W_{\mathbb Z^3}.$ Here $h_{\mathrm{sc}}$ is the simple-cubic spanning-tree entropy per vertex [3], Section 5.3, and $W_{\mathbb Z^3}$ is the simple-cubic Watson integral, normalized as the torus average of $1/(1-(\cos u+\cos v+\cos w)/3)$ [3], Section 3.

The coefficient is $k$ in the negative constant term $-k$. Changing $(x,y,z)$ to $(-x,-y,-z)$ proves $m(P_{-k})=m(P_k)$, so nonnegative parameters represent the whole real family. Zeros on the torus occur exactly for $0\leq k\leq6$ and are interpreted by the integrable logarithmic singularity, without a principal value or a modified measure.

The integer parameters $0,2,3,6,10,18$ are the six explicit CM examples in [1], Section 2.3. In this arithmetic terminology a singular K3 surface has Picard number $20$; this does not mean a zero of the defining polynomial on the integration torus.

This three-variable pencil extends the two-variable reciprocal diamond family to a K3 setting [1]. The two-variable table uses a positive constant term, which gives the same real-parameter values by its own sign symmetry.

Python

import mpmath as mp

def density(s):
    # A value assigned at an isolated integrable singularity changes no integral.
    if s == 0:
        return mp.mpf(0)
    return 1 / (4 * mp.pi * mp.agm(1, abs(s) / 4))


def compute(k, working):
    """Ordinary logarithmic measure for a finite real coefficient k."""
    with mp.workdps(working):
        k = abs(mp.mpf(k))
        if not mp.isfinite(k):
            raise ValueError('A finite real coefficient is required.')
        total = mp.mpf(0)
        pieces = [(-mp.mpf(4), min(mp.mpf(4), k - 2), 1),
                  (max(-mp.mpf(4), k + 2), mp.mpf(4), -1)]
        for left, right, sign in pieces:
            if left >= right:
                continue
            edges = [left] + ([mp.mpf(0)] if left < 0 < right else []) + [right]
            def integrand(s):
                t = sign * (k - s) - 2
                # acosh(1+t/2)=2asinh(sqrt(t)/2), stable at t=0.
                return density(s) * 2 * mp.asinh(mp.sqrt(max(0, t)) / 2)
            value, error = mp.quad(integrand, edges, error=True, maxdegree=10)
            if not mp.isfinite(value) or error > mp.power(10, -working + 20):
                raise ArithmeticError('Quadrature convergence estimate failed.')
            total += value
        return mp.nstr(total, working)


low = compute(7, 150)
high = compute(7, 210)
with mp.workdps(220):
    assert abs(mp.mpf(low)-mp.mpf(high)) < mp.mpf('1e-125')
    print(mp.nstr(mp.mpf(high), 100))
# Agreement checks are not certified bounds.

Marie-José Bertin, Amy Feaver, Jenny Fuselier, Matilde Lalín and Michelle Manes, Mahler measure of some singular K3-surfaces (2012). (arXiv)

Marie-José Bertin, Mesure de Mahler d'hypersurfaces K3 (2005). (arXiv)

Anthony J. Guttmann and Mathew D. Rogers, Spanning tree generating functions and Mahler measures, Journal of Physics A 45 (2012), 494001. (arXiv) (doi)

Mahler measure

Mahler_measures_of_x_x-1_y_y-1_k —   holds the two-variable reciprocal diamond family from which this K3 pencil is obtained by adding a third reciprocal pair

Mahler_measures_of_(short_random_walks) —   holds the three-step measure equal to $m(P_0)$

Values_of_Dirichlet_L-functions_at_positive_integers —   holds the quadratic Dirichlet value that evaluates $m(P_0)$

Entropy_constants_of_lattice_models —   holds the simple-cubic spanning-tree entropy equal to $m(P_6)$

Watson_integrals_of_the_cubic_lattices —   holds the simple-cubic endpoint derivative multiplied by six

Mahler measures of the Bertin $Q_k$ family —   holds Bertin's companion twelve-term K3 pencil with the same ordinary logarithmic convention; its special value $m(Q_{-3})=8m(P_0)/5$ uses the same three-step Mahler measure, while $m(Q_{12})$ is the face-centred cubic spanning-tree entropy corresponding to the simple-cubic value $m(P_6)$ here

Entries are of type: real number

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

Table is complete: no (it holds $k=0,1,\ldots,12$ and $18$; the symmetry center $0$ and torus-critical parameters $2,6$ are intrinsic, and $3,10,18$ complete the six integer CM examples displayed in [1], Section 2.3; the remaining integers $1,4,5,7,8,9,11,12$ are exploratory references for simple integral members in this fixed normalization, covering the interior and an exterior interval reaching twice the spectral edge $6$; the cutoff $12$ is a noncanonical practical choice independent of numeral notation, while $18$ adds arithmetic breadth without extending the whole slice; the attached argument-selection note records all groups and their rationale)

How they were obtained:

The generator evaluates the ordinary Jensen integral (1) by tanh-sinh quadrature at 150 and 210 working decimal digits, splitting at every logarithmic or square-root endpoint. The elliptic density is evaluated using the arithmetic-geometric mean to avoid subtractive loss near its singularity. All 14 entries agree to at least 149.66 digits relative to the larger of one and the value magnitude; 100 significant digits are retained. Quadrature estimates and precision agreement are numerical evidence, not certified error bounds.

more

An independent implementation evaluates the eta parametrization and divisor-sum expansion of [2], Theorem 1.1, at 140 working digits. It uses the six exact CM parameters of [1], Section 2.3, and solves the eta equation at the other arguments, recording the parameter residual and an explicit series-tail bound. It agrees with the quadrature to at least 139.58 digits on the same scale at every selected argument. The series-tail bound does not certify the numerical root solving or the full computation.

Independent controls give the density mass $1$ and second moment $4$, Smyth's value from Hurwitz zeta functions, and rejection of a deliberately perturbed value. Exact closed-walk coefficients and an explicit exterior tail bound reproduce every selected $k>6$ to at least 139.42 digits. The endpoint derivative agrees with Watson's gamma product to over 140 digits. The arithmetic relation (4), generator verification, and comparisons with the independent modular values are also checked on the stored decimal intervals. The attached checks and numerical evidence record the methods, software versions and measured errors.