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{
"Title": "Mahler measures of $(1+x)(1+y)(x+y)-k xy$",
"Definition": "The ordinary logarithmic Mahler measure CITE{RZ} 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": {
"type": "R",
"title": "coefficient parameter",
"display": "$k$",
"constraints": "$k\\in\\mathbb R$"
}
},
"Tags": [
"number theory",
"special values",
"elliptic curves",
"integral",
"statistical mechanics"
],
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "heuristic (agreement-checked)",
"complete": "no",
"complete-note": "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",
"rigour details": "The generator evaluates the real Jensen integral CITE{formula-jensen} 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.\n\nA 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.\n\nFurther 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."
},
"Display properties": {
"number-header": "$g(k)$"
},
"Numbers": {
"0": {
"number": "0",
"comment": "$P_0=(1+x)(1+y)(x+y)$, so $g(0)=0$ exactly."
},
"1": "0.2274812230123511078949823145226334852418855215224140089780860838723790203845983407353955022807987603",
"-1": {
"number": "0.6461318944389010281872730214476127881444836815611740322617369407220302256145396841675752181787427841",
"comment": "$P_{-1}=(x+y+1)(xy+x+y)$, so $g(-1)=2\\mu_3$ CITE{formula-special}."
},
"2": "0.4285945374649588653584255564520155133111245974494630836695954222555248414097766638980597225348640087",
"-2": "1.198701419402047373031779699852218640238111595312430159770119966120418925623670885901875555643418234",
"3": "0.6168709387895519462320052087397898437858587553166360837907332803671926343303332043034796762625528593",
"-3": "1.495911340204931681643419571997263500675864378521870912077004356073910000042052386490660240851970298",
"4": "0.7991342796013649153545197999014790934920743968749534398467466440802792837491139239345837037622788227",
"-4": "1.714378149859835461433702225808062053244498389797852334678381689022099365639106655592238890139456035",
"5": "0.9804186244483641768142619684084712615967093192188056886440029891454151266965433479333636299079908918",
"-5": "1.889400073010197519595004065656292956205863031612559019220069831066615496826984387782381921288052296",
"6": "1.166073129978612302274015858691141041137867387066762705860254441000064324928920282427965477886703096",
"-6": "2.036213383274183658614158807657091543473809198821210653153915195574739260040411823192582897130248665",
"7": "1.364887338074106647369893887135800911451313129134484053868516503234274122307590044412373013684792562",
"-7": "2.163020518434534504681833340403236959587630293006885090204320607652885964057232449003769439608410525",
"8": {
"number": "1.615329736097252570468182553619031970361209203902935080654342351805075564036349210418938045446856960",
"comment": "The triangular-lattice spanning-tree entropy, equal to twice the honeycomb-lattice spanning-tree entropy CITE{formula-special}."
},
"-8": "2.274812230123511078949823145226334852418855215224140089780860838723790203845983407353955022807987603",
"9": "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*sqrt(5)": {
"number": "1.744119984423508696227974482812483688784372090658490257546483620120659700186578289966398081804296897",
"param-latex": "$4+2\\sqrt5$"
},
"4-2*sqrt(5)": {
"number": "0.2537157145799035921583250169412140449458139015288933420703829900800385091862065198700611776014001594",
"param-latex": "$4-2\\sqrt5$"
},
"8+4*sqrt(5)": {
"number": "2.689105689245652477101429165723488284076669784442027075246220596161040116624042655998212459846314972",
"param-latex": "$8+4\\sqrt5$"
},
"8-4*sqrt(5)": {
"number": "0.5834057008831154620757395320381020072008931876343335114119613278404045307534017683889226971189570077",
"param-latex": "$8-4\\sqrt5$"
}
},
"Comments": {
"comment-convention": "Torus zeros are interpreted through the convergent logarithmic integral. They occur exactly for $-1\\leq k\\leq8$, and the cubic is singular exactly at $k=-1,0,8$ CITE{M}, Propositions 2.1 and 2.3. The sign of $k$ is retained: $g(k)$ is not an even function.",
"comment-family": "Boyd studied this reciprocal cubic alongside the HREF{Mahler_measures_of_x_x-1_y_y-1_k}[reciprocal diamond family] and the HREF{T649}[Hessian family] CITE{RZ}. The coefficient is $k$ itself here; the Hessian companion indexes its coefficient by its cube $t$."
},
"Formulas": {
"formula-jensen": "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 $y$.",
"formula-special": "$g(0)=0,\\quad g(-1)=2\\mu_3,\\quad g(8)=5\\mu_3=\\frac{15\\sqrt3}{4\\pi}L(2,\\chi_{-3})$, where $\\mu_3=m(1+x+y)$ is the HREF{Mahler_measures_of_(short_random_walks)#3}[three-step Mahler measure], and HREF{Values_of_Dirichlet_L-functions_at_positive_integers#3,2,2}[$L(2,\\chi_{-3})$] uses the character taking $0,1,-1$ on residues $0,1,2$ modulo $3$. The value $g(8)$ is the HREF{Entropy_constants_of_lattice_models#spanning-tree,triangular,entropy}[triangular spanning-tree entropy] and twice the HREF{Entropy_constants_of_lattice_models#spanning-tree,honeycomb,entropy}[honeycomb spanning-tree entropy] CITE{GR}, Section 5.1.",
"formula-elliptic": "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 $g(1)=\\frac7{2\\pi^2}L(f_{14},2)$, $g(7)=6g(1)$, $g(-8)=10g(1)$ CITE{M}; $g(4)=\\frac{10}{\\pi^2}L(f_{20},2)$, $g(-2)=\\frac32g(4)$, and $g(2)=\\frac9{2\\pi^2}L(f_{36},2)$ CITE{RZ}, Theorems 5 and 7 and the corollary to Theorem 5.",
"formula-quadratic": "$2g(4+2\\sqrt5)-g(8+4\\sqrt5)=g(4)=\\frac34N(32)$ CITE{RZ}, 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 HREF{T649}[Hessian family]. The two negative conjugate arguments are evaluated separately; this identity is stated for the positive embeddings."
},
"Similar tables": [
{
"table": "HREF{Mahler_measures_of_x_x-1_y_y-1_k}",
"relation": "holds the reciprocal diamond Mahler measures studied alongside this cubic family"
},
{
"table": "HREF{T649}",
"relation": "holds ordinary Hessian measures $N(t)$, with $g(4)=3N(32)/4$; its parameter is the cube of the coefficient"
},
{
"table": "HREF{Mahler_measures_of_(short_random_walks)}",
"relation": "holds $\\mu_3=m(1+x+y)$, with $g(-1)=2\\mu_3$ and $g(8)=5\\mu_3$"
},
{
"table": "HREF{Values_of_Dirichlet_L-functions_at_positive_integers}",
"relation": "holds $L(2,\\chi_{-3})$, which evaluates the singular-fibre values $g(-1)$ and $g(8)$"
},
{
"table": "HREF{Entropy_constants_of_lattice_models}",
"relation": "holds the triangular and honeycomb spanning-tree entropies $g(8)$ and $g(8)/2$"
}
],
"References": {
"RZ": {
"bib": "Mathew Rogers and Wadim Zudilin, From L-series of elliptic curves to Mahler measures, version 3 (2011).",
"arxiv": "1012.3036v3"
},
"M": {
"bib": "Anton Mellit, Elliptic dilogarithms and parallel lines (2012).",
"arxiv": "1207.4722v1"
},
"GR": {
"bib": "Anthony J. Guttmann and Mathew D. Rogers, Spanning tree generating functions and Mahler measures, Journal of Physics A 45 (2012), 494001.",
"arxiv": "1207.2815v2",
"doi": "10.1088/1751-8113/45/49/494001"
}
},
"Links": {
"RZonline": {
"title": "Rogers and Zudilin: definitions and arithmetic identities",
"url": "https://arxiv.org/html/1012.3036v3"
}
},
"Keywords": [
"Boyd",
"reciprocal cubic",
"Mahler measure",
"logarithmic Mahler measure",
"spanning tree",
"elliptic L-values"
],
"Programs": {
"program-python": {
"language": "Python",
"code": "# Ordinary reciprocal-cubic Mahler measure at one real argument.\n# Recompute radicals inside a callable, for example lambda: 4+2*mp.sqrt(5).\nimport mpmath as mp\n\ndef g(argument, working=140):\n with mp.workdps(working):\n k = mp.mpf(argument() if callable(argument) else argument)\n if not mp.isfinite(k):\n raise ValueError('A finite real argument is required.')\n if k == 0:\n return mp.mpf(0)\n edges = [mp.mpf(0), mp.pi/2]\n if -1 <= k < 0:\n edges += [mp.acos((1+s*mp.sqrt(1+k))/2) for s in [-1, 1]]\n elif 0 < k < 8:\n edges += [mp.acos((mp.sqrt(1+k)-1)/2)]\n edges = sorted(set(edges))\n pieces = []\n for left, right in zip(edges, edges[1:]):\n c = mp.cos((left+right)/2)\n active = abs(k-4*c*c) >= 4*c\n def integrand(theta):\n c = mp.cos(theta)\n if not active:\n return mp.log(2*c)\n b = abs(k-4*c*c)\n return mp.log((b+mp.sqrt(max(0, b*b-16*c*c)))/2)\n result, error = mp.quad(integrand, [left, right],\n error=True, maxdegree=9)\n if not mp.isfinite(result) or error >= mp.power(10, -working+15):\n raise ArithmeticError('Quadrature did not converge.')\n pieces.append(result)\n return 2*mp.fsum(pieces)/mp.pi\n\nlow = g(4, 140)\nhigh = g(4, 190)\nwith mp.workdps(200):\n assert abs(low-high) < mp.mpf('1e-115')*max(1, abs(high))\n print(mp.nstr(high, 100))\n# Precision agreement and quadrature estimates are not certified bounds.\n",
"comment": "This standalone mpmath routine evaluates one real argument. The example checks two working precisions before printing 100 significant digits. The attached generator enumerates all selected arguments."
}
}
}