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{
"Title": "Mahler measures of the Bertin $Q_k$ family",
"Definition": "The ordinary logarithmic Mahler measure CITE{Wiki} of Bertin's twelve-term family CITE{B}, $Q_k(x,y,z)=x+x^{-1}+y+y^{-1}+z+z^{-1}+xy+(xy)^{-1}+yz+(yz)^{-1}+xyz+(xyz)^{-1}-k$, 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 and Haar probability measure.",
"Parameters": {
"k": {
"type": "R",
"display": "$k$",
"title": "coefficient parameter",
"constraints": "$k\\in\\mathbb R$"
}
},
"Tags": [
"number theory",
"special values",
"integral",
"statistical mechanics",
"random walks"
],
"References": {
"B": {
"bib": "Marie-José Bertin, Mesure de Mahler d’hypersurfaces K3 (2005).",
"arxiv": "math/0501153"
},
"D": {
"bib": "Marie-José Bertin, A Mahler measure of a K3-hypersurface expressed as a Dirichlet L-series (2009).",
"arxiv": "0901.0619"
},
"BF": {
"bib": "Marie-José Bertin, Amy Feaver, Jenny Fuselier, Matilde Lalín and Michelle Manes, Mahler measure of some singular K3-surfaces (2012).",
"arxiv": "1208.6240"
},
"R": {
"bib": "Mathew D. Rogers, New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/pi (2008).",
"arxiv": "0704.2438v4"
},
"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"
},
"BSWZ": {
"bib": "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": "1103.2995",
"doi": "10.4153/CJM-2011-079-2"
}
},
"Links": {
"Wiki": {
"title": "Mahler measure",
"url": "https://en.wikipedia.org/wiki/Mahler_measure"
}
},
"Data properties": {
"type": "R",
"rigour": "heuristic (agreement-checked)",
"complete": "no",
"complete-note": "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",
"digits": "100",
"rigour details": "The generator uses Bertin's eta quotient and divisor-sum expansion CITE{B}, 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 CITE{formula-walk} 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).\n\nAn 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.\n\nExact 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."
},
"Display properties": {
"number-header": "$m(Q_k)$"
},
"Keywords": [
"Bertin",
"K3 surface",
"twelve-term Laurent polynomial",
"ordinary logarithmic Mahler measure",
"face-centred cubic",
"fcc",
"Domb numbers",
"four-step random walk",
"spanning-tree entropy"
],
"Comments": {
"comment-convention": "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.",
"comment-family": "This is the second of the two K3 pencils studied by Bertin CITE{B}; the HREF{Mahler_measures_of_x_x-1_y_y-1_z_z-1-k}[six-term pencil $P_k$] is its companion. The finite singular parameters of this $Q$ pencil are $-4,0,12$ CITE{B}, Section 3.1. A singular K3 surface in the arithmetic sense has Picard number $20$, which is a different use of the word CITE{BF}.",
"comment-random-walk": "Let $R_4$ be the distance from the origin after four independent planar steps of length $1$, each with uniformly distributed direction. The HREF{Moments_of_uniform_random_walks_in_the_plane}[radial moments] are $W_4(s)=\\mathbb E(R_4^s)$. The integer moments $D_n=W_4(2n)$ in CITE{formula-moments} are the Domb numbers CITE{BSWZ}."
},
"Formulas": {
"formula-walk": "$\\displaystyle Q_k(x,y,z)=\\frac{(1+x+xy+xyz)(1+z+zy+zyx)}{xyz}-(k+4),\\qquad m(Q_k)=\\mathbb E\\log|R_4^2-(k+4)|.$ 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 $R_4^2$.",
"formula-fcc": "$\\displaystyle Q_k(xz,y/z,z/x)=(x+x^{-1})(y+y^{-1})+(x+x^{-1})(z+z^{-1})+(y+y^{-1})(z+z^{-1})-k.$ This torus substitution preserves Haar probability measure. In the notation of CITE{R}, equation (1.2), $m(Q_k)=g_2(k+4)$.",
"formula-moments": "$\\displaystyle m(Q_k)=\\log|u|-\\sum_{n=1}^{\\infty}\\frac{D_n}{n u^n},\\qquad u=k+4,\\quad |u|>16,\\qquad D_n=\\sum_{j=0}^n\\binom nj^2\\binom{2j}j\\binom{2n-2j}{n-j}.$ These moments satisfy $D_0=1$, $D_1=4$ and $n^3D_n=2(2n-1)(5n^2-5n+2)D_{n-1}-64(n-1)^3D_{n-2}$ for $n\\geq2$ CITE{R}, equation (2.11), and CITE{BSWZ}, equation (37).",
"formula-special": "$\\displaystyle m(Q_{-4})=2m(1+x+y+z)=\\frac{7\\zeta(3)}{\\pi^2},\\qquad m(Q_{-3})=\\frac85m(1+x+y)=\\frac{6\\sqrt3}{5\\pi}L(2,\\chi_{-3}).$ The HREF{Mahler_measures_of_(short_random_walks)#4}[four-step measure] and the HREF{Mahler_measures_of_(short_random_walks)#3}[three-step measure] have the normalizations in CITE{BF}, Section 1. Here HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers#3}[$\\zeta(3)$] is Apéry's constant, 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 evaluation at $-4$ follows from CITE{formula-walk}; the evaluation at $-3$ is CITE{D}, Theorem 1.1.",
"formula-lattice": "$\\displaystyle m(Q_{12})=4m(Q_0)=h_{\\mathrm{fcc}},\\qquad 12\\left.\\frac{d}{dk}m(Q_k)\\right|_{k=12^+}=W_{A_3}.$ The constant $h_{\\mathrm{fcc}}$ is the face-centred cubic spanning-tree entropy per vertex CITE{GR}, Section 5.6; the factor-four relation is CITE{B}, Theorem 4.1. The HREF{Watson_integrals_of_the_cubic_lattices#A,3}[Watson integral $W_{A_3}$] is the torus average of $1/(1-(\\cos u\\cos v+\\cos v\\cos w+\\cos w\\cos u)/3)$."
},
"Similar tables": [
{
"table": "HREF{Mahler_measures_of_x_x-1_y_y-1_z_z-1-k}",
"relation": "holds Bertin's companion six-term K3 pencil, with the same negative-constant-term and logarithmic conventions"
},
{
"table": "HREF{Mahler_measures_of_(short_random_walks)}",
"relation": "holds the three- and four-step measures that evaluate $m(Q_{-3})$ and $m(Q_{-4})$"
},
{
"table": "HREF{Values_of_Dirichlet_L-functions_at_positive_integers}",
"relation": "holds $L(2,\\chi_{-3})$, which evaluates $m(Q_{-3})$"
},
{
"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}",
"relation": "holds $\\zeta(3)$, which evaluates $m(Q_{-4})$"
},
{
"table": "HREF{Moments_of_uniform_random_walks_in_the_plane}",
"relation": "holds radial moments of the same four-step random walk whose shifted logarithmic potential is $m(Q_k)$"
},
{
"table": "HREF{Watson_integrals_of_the_cubic_lattices}",
"relation": "holds the face-centred cubic endpoint derivative multiplied by twelve"
},
{
"table": "HREF{Entropy_constants_of_lattice_models}",
"relation": "holds spanning-tree entropies of other lattices; the face-centred cubic entropy occurs here at $k=12$"
}
],
"Programs": {
"program-exterior": {
"language": "Python",
"code": "# mpmath, ordinary measure for real k with abs(k+4)>16.\nimport mpmath as mp\n\ndef mahler_q_exterior(k, dps=120):\n with mp.workdps(dps):\n u = mp.mpf(k) + 4\n if abs(u) <= 16:\n raise ValueError(\"This series requires abs(k+4)>16\")\n r = 16 / abs(u)\n a, b, power = 1, 4, 1/u\n result = mp.log(abs(u)) - 4*power\n n = 1\n while r**(n+1)/((n+1)*(1-r)) > mp.eps:\n n += 1\n c = (2*(2*n-1)*(5*n*n-5*n+2)*b - 64*(n-1)**3*a)//n**3\n a, b = b, c\n power /= u\n result -= c*power/n\n return +result\n\nprint(mp.nstr(mahler_q_exterior(20), 100))"
}
},
"Numbers": {
"0": {
"number": "0.6032299748258814929381058622955367997707589321736622350967748119855432659481818994096147932919811545",
"comment": "The critical parameter $0$ satisfies $m(Q_{12})=4m(Q_0)$ CITE{B}, Theorem 4.1."
},
"1": {
"number": "0.8157244463298016021888138354733500673131506781203769679359984860077887345709016802656579317354979209"
},
"-1": {
"number": "0.4798982839841660702036502661261819545052697315982521828115657471045224126138360608517727218714492053"
},
"2": {
"number": "1.008934852559066769512004499960397133779898679136694526800141477411747735648446233713148865076590235"
},
"-2": {
"number": "0.4500037835560910960408547039038949026736869303779235239754652483676383165876885433168124691137731440"
},
"3": {
"number": "1.187129637726460481693555421863992848342603412960658388047855298092803210042892055852305261015957073"
},
"-3": {
"number": "0.5169055155511208225498184171580902305155869452489392258093895525776241804916317473340601745429942273",
"comment": "The Dirichlet evaluation $m(Q_{-3})=\\frac85m(1+x+y)$ is proved in CITE{D}, Theorem 1.1."
},
"4": {
"number": "1.353184954231269106150060654595395883532411255274121503198088564217949044461427480092749977792957654"
},
"-4": {
"number": "0.8525567976350115818470428531923337461160135392592702150832129160530589024582331629782483766644858872",
"comment": "$Q_{-4}=(1+x+xy+xyz)(1+z+zy+zyx)/(xyz)$, so $m(Q_{-4})=7\\zeta(3)/\\pi^2$ CITE{formula-special}."
},
"5": {
"number": "1.509152804444362335350895726639283717389850732765818610613280085324754006722940537844840579735011828"
},
"-5": {
"number": "1.372560366049330802276556142169595285936018127443409529843220547223643859983879937146678987439232602"
},
"6": {
"number": "1.656560445185824524582456040502851783966608361996713375051262633289788424409483561363149687865580040"
},
"-6": {
"number": "1.638774264006930172918454171953979237326960439275583451941359059198195989170196047420079536867235138",
"comment": "An arithmetic example studied together with $k=-36$ in CITE{B}, Theorem 4.2."
},
"7": {
"number": "1.796582289760286514836429268921603067975350187470638050150615540689583281655275424615599927182130799"
},
"-7": {
"number": "1.836576067953899936088319869429509498690673234565870151012616645128482107492585176240735337238885799"
},
"8": {
"number": "1.930145204086615494088169515978735403335836530584231440903835478945177060829195429684093523931880167"
},
"-8": {
"number": "1.996645441429820653628167361244861600331868680968162284467567833120295199236034909673389425312212551"
},
"9": {
"number": "2.057996363604673013850894877978010070606302447670700295734755941990609383096807812156689396926209207"
},
"10": {
"number": "2.180748812083957221521972263749029242656301411167314034307899347410958312362605806678176193562425969"
},
"11": {
"number": "2.298913101615050428821356972346335667305548106252146597768888683442695561189334226774286133712396204"
},
"12": {
"number": "2.412919899303525971752423449182147199083035728694648940387099247942173063792727597638459173167924618",
"comment": "The face-centred cubic spanning-tree entropy per vertex CITE{GR}, Section 5.6."
},
"13": {
"number": "2.510349823978290116046132391331902094040916542187368238450777364873612433534413572463700259283682665"
},
"14": {
"number": "2.594950989351357850625966134630110874953617557740066362954004636873710350447850302121855079253958035"
},
"15": {
"number": "2.671355074581672284985170448133965239402758378775511014041881810391297656531988496662501821068075483"
},
"16": {
"number": "2.741444090342137436657011798773635612507214384415309688303690519236098136553162709000312042331502930"
},
"-36": {
"number": "3.579163515426801092305961275055726874539300344637998021431105524389163611314483044544966470380460854",
"comment": "An arithmetic example studied together with $k=-6$ in CITE{B}, Theorem 4.2."
}
}
}