Mahler measures of $(1+x)(1+y)(x+y)-k xy$

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Numbers

$k$

$g(k)$

comment: $P_0=(1+x)(1+y)(x+y)$, so $g(0)=0$ exactly.

0.2274812230123511078949823145226334852418855215224140089780860838723790203845983407353955022807987603

-1:

0.6461318944389010281872730214476127881444836815611740322617369407220302256145396841675752181787427841

comment: $P_{-1}=(x+y+1)(xy+x+y)$, so $g(-1)=2\mu_3$ (2).

0.4285945374649588653584255564520155133111245974494630836695954222555248414097766638980597225348640087

-2:

1.198701419402047373031779699852218640238111595312430159770119966120418925623670885901875555643418234

0.6168709387895519462320052087397898437858587553166360837907332803671926343303332043034796762625528593

-3:

1.495911340204931681643419571997263500675864378521870912077004356073910000042052386490660240851970298

0.7991342796013649153545197999014790934920743968749534398467466440802792837491139239345837037622788227

-4:

1.714378149859835461433702225808062053244498389797852334678381689022099365639106655592238890139456035

0.9804186244483641768142619684084712615967093192188056886440029891454151266965433479333636299079908918

-5:

1.889400073010197519595004065656292956205863031612559019220069831066615496826984387782381921288052296

1.166073129978612302274015858691141041137867387066762705860254441000064324928920282427965477886703096

-6:

2.036213383274183658614158807657091543473809198821210653153915195574739260040411823192582897130248665

1.364887338074106647369893887135800911451313129134484053868516503234274122307590044412373013684792562

-7:

2.163020518434534504681833340403236959587630293006885090204320607652885964057232449003769439608410525

1.615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960

comment: The triangular-lattice spanning-tree entropy, equal to twice the honeycomb-lattice spanning-tree entropy (2).

-8:

2.274812230123511078949823145226334852418855215224140089780860838723790203845983407353955022807987603

1.850612816368655838696015626219369531357576265949908251372199841101577902990999612910439028787658578

-9:

2.374876904698177228601590074100619204168915741303179150807207645584569932968681118775905306229355069

10:

2.014023747519163689806197767464978251247687269923230815096926466878584448645512339285871054122156997

-10:

2.465507242647176725589822345437549136371909997838926659205663396463757503208932930938668713509488040

11:

2.148764337539612010325294392576878288914556276251051314280091568094815520420947998888564526246698732

-11:

2.548370992011640225488154913054985322952767262816007968199642032798384956121775243362227113451938554

12:

2.264998213493028189231941079367055833221164624317550764892708043368220139411037513795464687440566116

-12:

2.624722131983164000424985640075859139426784297832789872860687990867185066077367444092065397588484338

13:

2.367790014429108599317591349247137927936170905497804845016785456638532187217315955510339223941447443

-13:

2.695528961727592590537149356607496206382067448465671502602998644726578613524230055847858474769642554

14:

2.460203420276335898958237467125648519595046766385292066097849856089010268082745539586480339111896062

-14:

2.761555739733159012751059924971228190751358114804604638881458377695715481287113352931390422512914847

15:

2.544288799380897542354156261104728599959369401422023022074927135312703997218809137422634200967141737

-15:

2.823416856207120375397604528760164315230913512644095349066302954387795499275477351159522380059808606

16:

2.621508037963306413788168382615452908953099685415509466456029312373304208553271915466126392977738448

-16:

2.881614000809653114009940950531690284105816508319603719325098859282740251166332817271386825709643908

$4+2\sqrt5$:

1.744119984423508696227974482812483688784372090658490257546483620120659700186578289966398081804296897

$4-2\sqrt5$:

0.2537157145799035921583250169412140449458139015288933420703829900800385091862065198700611776014001594

$8+4\sqrt5$:

2.689105689245652477101429165723488284076669784442027075246220596161040116624042655998212459846314972

$8-4\sqrt5$:

0.5834057008831154620757395320381020072008931876343335114119613278404045307534017683889226971189570077

Definition

The ordinary logarithmic Mahler measure [1] of $P_k(x,y)=(1+x)(1+y)(x+y)-kxy$ is $g(k)=\int_0^1\int_0^1\log|P_k(e^{2\pi i u},e^{2\pi i v})|\,du\,dv$ for real $k$, using natural logarithms and normalized Haar measure.

Parameters

$k$

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

Formulas

(1)

With $c=\cos\theta$ and $b=|k-4c^2|$, Jensen's formula gives $g(k)=\frac2\pi\int_0^{\pi/2}J_k(\theta)\,d\theta$, where $J_k(\theta)=\begin{cases}\log(2c),&b\leq4c,\\\log\bigl((b+\sqrt{b^2-16c^2})/2\bigr),&b>4c.\end{cases}$ Endpoint values are interpreted by limits. This follows by treating $P_k(e^{2i\theta},y)$ as a quadratic in $y$.

(2)

$g(0)=0,\quad g(-1)=2\mu_3,\quad g(8)=5\mu_3=\frac{15\sqrt3}{4\pi}L(2,\chi_{-3})$, where $\mu_3=m(1+x+y)$ is the three-step Mahler measure, and $L(2,\chi_{-3})$ uses the character taking $0,1,-1$ on residues $0,1,2$ modulo $3$. The value $g(8)$ is the triangular spanning-tree entropy and twice the honeycomb spanning-tree entropy [3], Section 5.1.

(3)

Let $\eta(\tau)=q^{1/24}\prod_{n\geq1}(1-q^n)$ with $q=e^{2\pi i\tau}$. Put $f_{14}=\eta(\tau)\eta(2\tau)\eta(7\tau)\eta(14\tau)$, $f_{20}=\eta(2\tau)^2\eta(10\tau)^2$, and $f_{36}=\eta(6\tau)^4$. For $f=\sum_{n\geq1}a_nq^n$, write $L(f,s)=\sum_{n\geq1}a_nn^{-s}$. Then $g(1)=\frac7{2\pi^2}L(f_{14},2)$, $g(7)=6g(1)$, $g(-8)=10g(1)$ [2]; $g(4)=\frac{10}{\pi^2}L(f_{20},2)$, $g(-2)=\frac32g(4)$, and $g(2)=\frac9{2\pi^2}L(f_{36},2)$ [1], Theorems 5 and 7 and the corollary to Theorem 5.

(4)

$2g(4+2\sqrt5)-g(8+4\sqrt5)=g(4)=\frac34N(32)$ [1], Theorem 5, where $\sqrt5$ is positive and $N(t)=m(x^3+y^3+1-\sqrt[3]{t}xy)$ uses the real cube root as in the Hessian family. The two negative conjugate arguments are evaluated separately; this identity is stated for the positive embeddings.

Comments

(5)

Torus zeros are interpreted through the convergent logarithmic integral. They occur exactly for $-1\leq k\leq8$, and the cubic is singular exactly at $k=-1,0,8$ [2], Propositions 2.1 and 2.3. The sign of $k$ is retained: $g(k)$ is not an even function.

(6)

Boyd studied this reciprocal cubic alongside the reciprocal diamond family and the Hessian family [1]. The coefficient is $k$ itself here; the Hessian companion indexes its coefficient by its cube $t$.

Programs

(P1)

Python

# Ordinary reciprocal-cubic Mahler measure at one real argument.
# Recompute radicals inside a callable, for example lambda: 4+2*mp.sqrt(5).
import mpmath as mp

def g(argument, working=140):
    with mp.workdps(working):
        k = mp.mpf(argument() if callable(argument) else argument)
        if not mp.isfinite(k):
            raise ValueError('A finite real argument is required.')
        if k == 0:
            return mp.mpf(0)
        edges = [mp.mpf(0), mp.pi/2]
        if -1 <= k < 0:
            edges += [mp.acos((1+s*mp.sqrt(1+k))/2) for s in [-1, 1]]
        elif 0 < k < 8:
            edges += [mp.acos((mp.sqrt(1+k)-1)/2)]
        edges = sorted(set(edges))
        pieces = []
        for left, right in zip(edges, edges[1:]):
            c = mp.cos((left+right)/2)
            active = abs(k-4*c*c) >= 4*c
            def integrand(theta):
                c = mp.cos(theta)
                if not active:
                    return mp.log(2*c)
                b = abs(k-4*c*c)
                return mp.log((b+mp.sqrt(max(0, b*b-16*c*c)))/2)
            result, error = mp.quad(integrand, [left, right],
                                    error=True, maxdegree=9)
            if not mp.isfinite(result) or error >= mp.power(10, -working+15):
                raise ArithmeticError('Quadrature did not converge.')
            pieces.append(result)
        return 2*mp.fsum(pieces)/mp.pi

low = g(4, 140)
high = g(4, 190)
with mp.workdps(200):
    assert abs(low-high) < mp.mpf('1e-115')*max(1, abs(high))
    print(mp.nstr(high, 100))
# Precision agreement and quadrature estimates are not certified bounds.

References

[1]

Mathew Rogers and Wadim Zudilin, From L-series of elliptic curves to Mahler measures, version 3 (2011). (arXiv)

[2]

Anton Mellit, Elliptic dilogarithms and parallel lines (2012). (arXiv)

[3]

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

Links

[4]

Rogers and Zudilin: definitions and arithmetic identities

Similar tables

Mahler_measures_of_x_x-1_y_y-1_k — holds the reciprocal diamond Mahler measures studied alongside this cubic family

T649 — holds ordinary Hessian measures $N(t)$, including $N(32)$ with $g(4)=3N(32)/4$; its parameter is the cube of the coefficient

Mahler_measures_of_(short_random_walks) — holds $\mu_3=m(1+x+y)$, with $g(-1)=2\mu_3$ and $g(8)=5\mu_3$

Values_of_Dirichlet_L-functions_at_positive_integers — holds $L(2,\chi_{-3})$, which evaluates the singular-fibre values $g(-1)$ and $g(8)$

Entropy_constants_of_lattice_models — holds the triangular and honeycomb spanning-tree entropies $g(8)$ and $g(8)/2$

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 $-16\leq k\leq16$ and both real embeddings of $4+2\sqrt5$ and $8+4\sqrt5$; the singular fibres $-1,0,8$, the published arithmetic cases $-8,-2,1,2,4,7,4+2\sqrt5,8+4\sqrt5$, and the two conjugate embeddings have distinct structural or arithmetic roles; the remaining 24 integers are exploratory references for small integral polynomials in this fixed normalization, covering both exterior regimes and the torus-zero interval; the absolute coefficient bound 16 is a noncanonical practical choice, independent of numeral notation, that extends beyond the boundary at 8 while treating negative inputs equally; the quadratic pairs add arithmetic breadth without an additional mesh; the attached argument-selection note records all memberships and sources)

How they were obtained:

The generator evaluates the real Jensen integral (1) by adaptive tanh-sinh quadrature at 140 and 190 working decimal digits. Every crossing $|k-4\cos^2\theta|=4\cos\theta$ is an integration breakpoint. Radical arguments are recomputed at each precision. All 36 nonzero entries agree to at least 139 digits relative to the larger of one and the value magnitude; 100 significant digits are retained. The quadrature convergence estimates and agreement checks are not certified error bounds. The entry $k=0$ is exact by factorization.

A separate implementation integrates the logarithms of the root moduli of $P_k(e^{i\theta},y)$ directly at 125 working digits and agrees for all 37 arguments to more than 110 digits. It first reproduces the nonzero control $g(-1)=2\mu_3$, with $\mu_3$ independently evaluated by Hurwitz zeta functions. The direct-root check shares Jensen's principle with the generator but uses the original complex polynomial.

Further checks use Rogers and Zudilin's equation (7) only outside its excluded interval $[-4,2]$, the exact factorizations at $k=0,-1$, the spanning-tree evaluation at $k=8$, PARI elliptic-curve $L$-values of conductors 14, 20 and 36, and the positive quadratic identity. The elliptic models have the stated conductors and match the cited eta products through $q^{80}$. A deliberately perturbed value is rejected. The attached checks and numerical evidence record the computations and comparisons with values read back from the database.