Mahler measures of the Bertin $Q_k$ family

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Numbers

$k$

$m(Q_k)$

0.6032299748258814929381058622955367997707589321736622350967748119855432659481818994096147932919811545

comment: The critical parameter $0$ satisfies $m(Q_{12})=4m(Q_0)$ [1], Theorem 4.1.

0.8157244463298016021888138354733500673131506781203769679359984860077887345709016802656579317354979209

-1:

0.4798982839841660702036502661261819545052697315982521828115657471045224126138360608517727218714492053

1.008934852559066769512004499960397133779898679136694526800141477411747735648446233713148865076590235

-2:

0.4500037835560910960408547039038949026736869303779235239754652483676383165876885433168124691137731440

1.187129637726460481693555421863992848342603412960658388047855298092803210042892055852305261015957073

-3:

0.5169055155511208225498184171580902305155869452489392258093895525776241804916317473340601745429942273

comment: The Dirichlet evaluation $m(Q_{-3})=\frac85m(1+x+y)$ is proved in [2], Theorem 1.1.

1.353184954231269106150060654595395883532411255274121503198088564217949044461427480092749977792957654

-4:

0.8525567976350115818470428531923337461160135392592702150832129160530589024582331629782483766644858872

comment: $Q_{-4}=(1+x+xy+xyz)(1+z+zy+zyx)/(xyz)$, so $m(Q_{-4})=7\zeta(3)/\pi^2$ (4).

1.509152804444362335350895726639283717389850732765818610613280085324754006722940537844840579735011828

-5:

1.372560366049330802276556142169595285936018127443409529843220547223643859983879937146678987439232602

1.656560445185824524582456040502851783966608361996713375051262633289788424409483561363149687865580040

-6:

1.638774264006930172918454171953979237326960439275583451941359059198195989170196047420079536867235138

comment: An arithmetic example studied together with $k=-36$ in [1], Theorem 4.2.

1.796582289760286514836429268921603067975350187470638050150615540689583281655275424615599927182130799

-7:

1.836576067953899936088319869429509498690673234565870151012616645128482107492585176240735337238885799

1.930145204086615494088169515978735403335836530584231440903835478945177060829195429684093523931880167

-8:

1.996645441429820653628167361244861600331868680968162284467567833120295199236034909673389425312212551

2.057996363604673013850894877978010070606302447670700295734755941990609383096807812156689396926209207

10:

2.180748812083957221521972263749029242656301411167314034307899347410958312362605806678176193562425969

11:

2.298913101615050428821356972346335667305548106252146597768888683442695561189334226774286133712396204

12:

2.412919899303525971752423449182147199083035728694648940387099247942173063792727597638459173167924618

comment: The face-centred cubic spanning-tree entropy per vertex [5], Section 5.6.

13:

2.510349823978290116046132391331902094040916542187368238450777364873612433534413572463700259283682665

14:

2.594950989351357850625966134630110874953617557740066362954004636873710350447850302121855079253958035

15:

2.671355074581672284985170448133965239402758378775511014041881810391297656531988496662501821068075483

16:

2.741444090342137436657011798773635612507214384415309688303690519236098136553162709000312042331502930

-36:

3.579163515426801092305961275055726874539300344637998021431105524389163611314483044544966470380460854

comment: An arithmetic example studied together with $k=-6$ in [1], Theorem 4.2.

Definition

The ordinary logarithmic Mahler measure [7] of Bertin's family [1], $\begin{aligned}Q_k(x,y,z)={}&x+x^{-1}+y+y^{-1}+z+z^{-1}\\&+xy+(xy)^{-1}+yz+(yz)^{-1}+xyz+(xyz)^{-1}-k\end{aligned}$, is $m(Q_k)=\int_{[0,1]^3}\log|Q_k(e^{2\pi i u},e^{2\pi i v},e^{2\pi i w})|\,du\,dv\,dw$, with natural logarithms.

Parameters

$k$

— coefficient parameter ($k\in\mathbb R$)

Formulas

(1)

$\begin{gathered}Q_k(x,y,z)=\frac{(1+x+xy+xyz)(1+z+zy+zyx)}{xyz}-(k+4),\\ m(Q_k)=\mathbb E\log|R_4^2-(k+4)|.\end{gathered}$ On the unit torus, the quotient is $|1+x+xy+xyz|^2$; the substitution $(U,V,W)=(x,xy,xyz)$ preserves Haar probability measure. Rotational invariance identifies this distribution with $R_4^2$.

(2)

$\begin{aligned}Q_k(xz,y/z,z/x)={}&(x+x^{-1})(y+y^{-1})\\&+(x+x^{-1})(z+z^{-1})\\&+(y+y^{-1})(z+z^{-1})-k.\end{aligned}$ This torus substitution preserves Haar probability measure. In the notation of [4], equation (1.2), $m(Q_k)=g_2(k+4)$.

(3)

$\begin{gathered}m(Q_k)=\log|u|-\sum_{n=1}^{\infty}\frac{D_n}{n u^n},\\ u=k+4,\quad |u|>16,\\ D_n=\sum_{j=0}^n\binom nj^2\binom{2j}j\binom{2n-2j}{n-j}.\end{gathered}$ These moments satisfy $D_0=1$, $D_1=4$ and $\begin{aligned}n^3D_n={}&2(2n-1)(5n^2-5n+2)D_{n-1}\\&-64(n-1)^3D_{n-2}\end{aligned}$ for $n\geq2$ [4], equation (2.11), and [6], equation (37).

(4)

$\begin{gathered}m(Q_{-4})=2m(1+x+y+z)=\frac{7\zeta(3)}{\pi^2},\\ m(Q_{-3})=\frac85m(1+x+y)=\frac{6\sqrt3}{5\pi}L(2,\chi_{-3}).\end{gathered}$ The four-step measure and the three-step measure have the normalizations in [3], Section 1. Here $\zeta(3)$ is Apéry's constant, and $L(2,\chi_{-3})$ uses the character taking $0,1,-1$ on residues $0,1,2$ modulo $3$. The evaluation at $-4$ follows from (1); the evaluation at $-3$ is [2], Theorem 1.1.

(5)

$\begin{gathered}m(Q_{12})=4m(Q_0)=h_{\mathrm{fcc}},\\ 12\left.\frac{d}{dk}m(Q_k)\right|_{k=12^+}=W_{A_3}.\end{gathered}$ The constant $h_{\mathrm{fcc}}$ is the face-centred cubic spanning-tree entropy per vertex [5], Section 5.6; the factor-four relation is [1], Theorem 4.1. The Watson integral $W_{A_3}$ is the torus average of $1/(1-(\cos u\cos v+\cos v\cos w+\cos w\cos u)/3)$.

Comments

(6)

The coefficient is $k$ in the negative constant term $-k$. Its sign is retained: $m(Q_k)$ is not an even function. Torus zeros occur exactly for $-4\leq k\leq12$ and are treated by the integrable logarithmic singularity, without a principal value or a modified measure.

(7)

This is the second of the two K3 pencils studied by Bertin [1]; the six-term pencil $P_k$ is its companion. The finite singular parameters of this $Q$ pencil are $-4,0,12$ [1], Section 3.1. A singular K3 surface in the arithmetic sense has Picard number $20$, which is a different use of the word [3].

(8)

Let $R_4$ be the distance from the origin after four independent planar steps of length $1$, each with uniformly distributed direction. The radial moments are $W_4(s)=\mathbb E(R_4^s)$. The integer moments $D_n=W_4(2n)$ in (3) are the Domb numbers [6].

Programs

(P1)

Python

# mpmath, ordinary measure for real k with abs(k+4)>16.
import mpmath as mp

def mahler_q_exterior(k, dps=120):
    with mp.workdps(dps):
        u = mp.mpf(k) + 4
        if abs(u) <= 16:
            raise ValueError("This series requires abs(k+4)>16")
        r = 16 / abs(u)
        a, b, power = 1, 4, 1/u
        result = mp.log(abs(u)) - 4*power
        n = 1
        while r**(n+1)/((n+1)*(1-r)) > mp.eps:
            n += 1
            c = (2*(2*n-1)*(5*n*n-5*n+2)*b - 64*(n-1)**3*a)//n**3
            a, b = b, c
            power /= u
            result -= c*power/n
        return +result

print(mp.nstr(mahler_q_exterior(20), 100))

References

[1]

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

[2]

Marie-José Bertin, A Mahler measure of a K3-hypersurface expressed as a Dirichlet L-series (2009). (arXiv)

[3]

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

[4]

Mathew D. Rogers, New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/pi (2008). (arXiv)

[5]

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

[6]

Jonathan M. Borwein, Armin Straub, James Wan and Wadim Zudilin, Densities of short uniform random walks, Canadian Journal of Mathematics 64 (2012), 961–990, with an appendix by Don Zagier. (arXiv) (doi)

Links

[7]

Mahler measure

Similar tables

Mahler measures of Bertin's six-term pencil $P_k$ — holds Bertin's companion six-term K3 pencil, with the same negative-constant-term and logarithmic conventions

Mahler measures of uniform planar random walks — holds the three- and four-step measures that evaluate $m(Q_{-3})$ and $m(Q_{-4})$

Dirichlet $L$-values at positive integers — holds $L(2,\chi_{-3})$, which evaluates $m(Q_{-3})$

Riemann zeta values at rational arguments — holds $\zeta(3)$, which evaluates $m(Q_{-4})$

Moments of uniform random walks in the plane — holds radial moments of the same four-step random walk whose shifted logarithmic potential is $m(Q_k)$

Watson integrals of the cubic lattices — holds the face-centred cubic endpoint derivative multiplied by twelve

Entropy constants of lattice models — holds spanning-tree entropies of other lattices; the face-centred cubic entropy occurs here at $k=12$

Data properties

Entries are of type: real number

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

Table is complete: no (it holds every integer $-8\leq k\leq16$ and $k=-36$; the singular parameters $-4,0,12$ are intrinsic, and $-3,-6,-36$ are cited arithmetic examples; the other integers provide exploratory references for simple integral polynomials in this normalization, covering the torus-zero interval and four coefficients into each exterior regime; the exterior bounds are noncanonical practical choices independent of numeral notation, while $-36$ adds arithmetic breadth without filling a deeper interval; the attached argument-selection note gives the exhaustive groups and checked sources)

How they were obtained:

The generator uses Bertin's eta quotient and divisor-sum expansion [1], Theorem 1.1(2), with explicit branch choices for the nome. It reconstructs the exact CM nomes where available and solves the eta equation afresh at 150 and 210 working decimal digits elsewhere. The endpoint $k=-4$ uses the factorization in (1) and Smyth's evaluation. Every entry agrees between the two precisions to at least 149.33 digits relative to the larger of one and the value magnitude; 100 significant digits are retained. The omitted divisor-sum tail is explicitly bounded, but the root solving and floating-point arithmetic are not certified, so the common rigour level is heuristic (agreement-checked).

An independent implementation starts with the Domb-moment series in its convergence region and continues its differential equation through the upper half-plane to each of the 23 nonsingular selected arguments. Its known-answer control continues from $u=32$ to $u=64$ and agrees with a direct exterior sum. At $k=0,12$, a separate real hypergeometric endpoint integral and the proved factor-four relation provide the check. These independent values agree with the generator to at least 139.32 absolute decimal digits. Arb separately evaluates the zeta and Dirichlet formulas at $k=-4,-3$, agreeing to more than 185 digits.

Exact exponent arithmetic verifies both torus substitutions; direct binomial sums verify the first 35 Domb coefficients. The exterior series is also summed directly with an explicit tail bound for every selected $|k+4|>16$. The endpoint derivative is checked against Watson's gamma product. A deliberately perturbed value is rejected. Generator verification and the same independent comparisons are applied to the stored decimal intervals. The attached checks, argument-selection note and numerical evidence record the computations and software versions.