Mahler measures of $1+x_1+\dots+x_{n-1}$ (short random walks)

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Numbers

$n$

$\mu_n$

0.32306594721945051409363651072380639407224184078059

comment: Smyth's value $\mu_3=\mathrm{Cl}_2(\pi/3)/\pi=\frac{3\sqrt3}{4\pi}L(2,\chi_{-3})$.

0.42627839881750579092352142659616687305800676962964

comment: Smyth's value $\mu_4=7\zeta(3)/(2\pi^2)$.

0.54441256175218558519587806274502767666605280202853

comment: The value is conjecturally equal to the Rodriguez Villegas eta-integral (6); Borwein, Straub, Wan and Zudilin confirmed this numerically to 600 decimal places [1].

0.62731707483690980718358664940461715251475544677472

comment: The value is conjecturally equal to the Rodriguez Villegas eta-integral (7); Borwein, Straub, Wan and Zudilin confirmed this numerically to 80 decimal places [1].

0.70292629247696726678782394439522698211409870734036

0.76683108806961275701405175616677786312599958330277

0.82415623953238869482052282484961993900965218979887

10:

0.87532865811448453408495821795044968468699477558650

11:

0.92185088673265369756589152797032793838897337911216

12:

0.96437578931955973661558078511794110995312118915484

13:

1.0035835304893201106044538743208630678161512767344

14:

1.0399353083414942797155439822506320153315071867840

15:

1.0738262172568560361842527815003012679261311018007

16:

1.1055653432007483833873406105210686981784482021274

17:

1.1354107037674110729532392500429850075491579196068

18:

1.1635750742159058991157464496624842897920815280862

19:

1.1902378646444420543605782001962175319482443395698

20:

1.2155509916488112621332092556227863885614917915569

21:

1.2396445218138194484770407126508737950376274991631

22:

1.2626305503381166068793048205526759349514404819086

23:

1.2846064131526907910836576958870754556021479068505

24:

1.3056571499292296515274590218273370827741209593247

25:

1.3258574988525685385706252559404011448100268445135

26:

1.3452734927753255590497508719516495351310169955952

27:

1.3639637617393380351637983352681897157816417357688

28:

1.3819805991129638675001940124766754886249700346555

29:

1.3993708431121593694523323205727774721323322361479

30:

1.4161766099427499297354865816196209045424106070005

31:

1.4324359078908890337931505693047475891528597246144

32:

1.4481831546957277096734328364120036982525480599618

33:

1.4634496159778495854154822260927512130854408369597

34:

1.4782637787283674659437376950495542525318124825437

35:

1.4926516710770066812975632617733973346779678373610

36:

1.5066371373484661534152801073191897819987005019009

37:

1.5202420757062095912909461177595548193669763309686

38:

1.5334866443277113996544507264378061969532435976464

39:

1.5463894409826094883391187704089468458151920562272

40:

1.5589676600272793803012481004307138221734828353130

41:

1.5712372301402387391681312150663472454955683459898

42:

1.5832129355655052469811124571910609417849397128012

43:

1.5949085231780737178042936100786624659362516474088

44:

1.6063367973154949333751188621688885006599116610884

45:

1.6175097040155269724108525748334791433185285445952

46:

1.6284384060489602281703396858705023899322620805458

47:

1.6391333499287685552470046295484924314846848077895

48:

1.6496043259036208833517945791408319001778054597005

49:

1.6598605217990725700701169336350827182346327993675

50:

1.6699105714483063467267316894143229748693239866594

51:

1.6797625983519841401366461862922361452969831569130

52:

1.6894242551202751652337664526593024346877405367260

53:

1.6989027591767451851766503480759962860339280700598

54:

1.7082049251413311027626440223777756459493167661282

55:

1.7173371942562894085596569737942400522004321628408

56:

1.7263056611733236356302837507734212439516123247290

57:

1.7351160983808515055087582504381007820977589120454

58:

1.7437739785165636046539676784189445123379306519329

59:

1.7522844947812196552226749473601664736216470735638

60:

1.7606525796443325022190426736327804398223611581351

61:

1.7688829220104246873621710385928855138995585096713

62:

1.7769799829954225473134370111645874453583262589774

63:

1.7849480104460701264477415178198785641001865859628

64:

1.7927910523206557935149394379685661302108161310023

65:

1.8005129690365573839382504249572776645745677593196

66:

1.8081174448788802805808241116930857136766505599306

67:

1.8156079985545773707188906200495053976739178379679

68:

1.8229879929677215022333374766349317201537196141077

69:

1.8302606442838971341320760831563496788936646623780

70:

1.8374290303448574620683286375719747707609659330562

71:

1.8444960984885440208182758915963898321491080427379

72:

1.8514646728241909054987901331465405444235408920492

73:

1.8583374610074518872999206101235515562617979661081

74:

1.8651170605562237138052274320938723554970377180543

75:

1.8718059647440303109634820484260795698171712930211

76:

1.8784065681044262214967433467614965901693898573283

77:

1.8849211715768262650242907445789860786319663244516

78:

1.8913519873214309951991846791908086648476030608350

79:

1.8977011432284581831121446012360975580739508785840

80:

1.9039706871446778966010225340480115906680704628379

81:

1.9101625908382552797901589949648221305931274873127

82:

1.9162787537211067366892231733553998781680821904537

83:

1.9223210063463506732097882520155824085879654009600

84:

1.9282911136969645724822607544397392681134874058858

85:

1.9341907782804294919107910384280193510609642819524

86:

1.9400216430429365213760717041107857439899603608018

87:

1.9457852941156344525161119308017624959005056620336

88:

1.9514832634044024686607461568530771151690547003255

89:

1.9571170310337259489261579707441347228831401816480

90:

1.9626880276544284913810382522549029455039239056350

91:

1.9681976366242609935768892704432463610771703629803

92:

1.9736471960696619510631712300072532268363755262180

93:

1.9790380008363756817606088474370722101530645888600

94:

1.9843713043360412710024824054372142399190255617855

95:

1.9896483202953395734185145843747453307118773759329

96:

1.9948702244138040494356095611142246645059306016668

97:

2.0000381559359594724980993474508342269132061900544

98:

2.0051532191430469523848313504687600569604841378894

99:

2.0102164847692209856344789986043907688587956684306

100:

2.0152289913467614017614054613443826813755324569771

101:

2.0201917464845274535379376304670373866793475307995

102:

2.0251057280835904934245854598135941609707427140379

103:

2.0299718854937135128508924527541378155436962386002

104:

2.0347911406140983325619179291634842321388644293875

105:

2.0395643889415926442011142702129397794328863709892

106:

2.0442925005693378072284850501865042639414333884746

107:

2.0489763211386428433708656602420937929754477464915

108:

2.0536166727466891205543829096032373783294372612995

109:

2.0582143548125025787674640635265955897296812938049

110:

2.0627701449034749300529538629203309414366312000531

111:

2.0672847995245710708866541289378380298641354116258

112:

2.0717590548722260734733820960756484886282443273902

113:

2.0761936275548107490688475911975903000687156645115

114:

2.0805892152814291498331451576710257921103936666599

115:

2.0849464975207038099420967428067384234945356057424

116:

2.0892661361311043949915287645222535209161767083210

117:

2.0935487759642821580717211446122792081419930746257

118:

2.0977950454427856668746499300241092902756225143461

119:

2.1020055571134521885925011440870850479903772944292

120:

2.1061809081776934580407993156066181830633410617535

121:

2.1103216809998239057102458760624146695849309762583

122:

2.1144284435945134157902849921974562129611470209802

123:

2.1185017500943849792711714874390270711697822366557

124:

2.1225421411987198911780080845231842613177093867062

125:

2.1265501446041791260545090452223338017745975714732

126:

2.1305262754183989471138707794778744869333766418492

127:

2.1344710365572714180240700178414139646068720498814

128:

2.1383849191266760672080955789693352942108812677987

129:

2.1422684027893872953953197361002980099702146129242

130:

2.1461219561178430265233269035740777593797468280405

131:

2.1499460369334234031657455104485509903562322817086

132:

2.1537410926328538570751484947863749861866351565226

133:

2.1575075605023144921116963316052419135774464807914

134:

2.1612458680198072610032167486567159921319749693334

135:

2.1649564331463037696487259730743139649129762121032

136:

2.1686396646061695831649594664533562516622528740374

137:

2.1722959621573355254796182323012143226003424666815

138:

2.1759257168516625559565005087810132117024637975209

139:

2.1795293112859242766832564998053915238771285452967

140:

2.1831071198438098838904838917963166083057172722151

141:

2.1866595089293303440364448836148476887962007901378

142:

2.1901868371919916727374585565173628762741077439343

143:

2.1936894557440813516718231218812917672197621004238

144:

2.1971677083703970685151115512205318617505333889604

145:

2.2006219317307310461385767388373199396019821109824

146:

2.2040524555554081822177015140799487145967651899622

147:

2.2074596028341619954704304883746571478752489496746

148:

2.2108436899986189200167052502695617036357493660225

149:

2.2142050270986487582233798543181750357798040411403

150:

2.2175439179728270513839282145583187760307818783034

151:

2.2208606604132437160777463283618234017906684779277

152:

2.2241555463248814841320496213341441131499987889155

153:

2.2274288618797774403260195198645632086072779608037

154:

2.2306808876661712411928813420798176433564646110242

155:

2.2339118988328343894933232033292310473435598803513

156:

2.2371221652287662031455909233176892540953284469394

157:

2.2403119515384338274467299334875430813001933515416

158:

2.2434815174127257698702857044618163327922097735883

159:

2.2466311175957809637450130725354942785943737453811

160:

2.2497610020478482683653550051588238995864135729326

161:

2.2528714160643245676059992487091929156686323684327

162:

2.2559626003911132172533545953395829850477873969022

163:

2.2590347913364384945776922402336832490296420461286

164:

2.2620882208792459048289657856764453407301305223047

165:

2.2651231167743126820763391690197830313527790428716

166:

2.2681397026541875708372139438407813332995007605717

167:

2.2711381981280739758839174532700821935461161016660

168:

2.2741188188777658069597949552819391570709333867700

169:

2.2770817767507408101667414250287611194140764362701

170:

2.2800272798505118565377106915406617320238892245952

171:

2.2829555326243325395156965901904603043102318936566

172:

2.2858667359483495061162069176556889727100256241776

173:

2.2887610872102902014585662438637810229037474994952

174:

2.2916387803897711336929363874687762782354034757509

175:

2.2945000061363083572442019732366055716410237750062

176:

2.2973449518451086183655854002728327532475481210468

177:

2.3001738017307165003426216424558925285764412900978

178:

2.3029867368985899388550451498870984543470200834218

179:

2.3057839354146736439504867101248432386485261517009

180:

2.3085655723730372571615487027577791572884515138488

181:

2.3113318199616424842265238583414517627804136087355

182:

2.3140828475263009697189865876145980184573502618267

183:

2.3168188216328823140427002194389541059468111304680

184:

2.3195399061278293704010330121866827997355203209320

185:

2.3222462621970357944867700979214794106112952992127

186:

2.3249380484231387480113786426861108828484806638128

187:

2.3276154208412776743091715778826686813846819436417

188:

2.3302785329933681658574808595786241116116201918141

189:

2.3329275359809381256203246572988274294314631100101

190:

2.3355625785165716828338704607987238908618843259284

191:

2.3381838069740046555910572468185694793646752105022

192:

2.3407913654369137539224304180301697708358216145139

193:

2.3433853957464401847607732795926463967812168732813

194:

2.3459660375474868511364269783549707230266330995549

195:

2.3485334283338269292544706750568366923677000659337

196:

2.3510877034920602559797409316262971397934148527944

197:

2.3536289963444526630675711771946796217575530386436

198:

2.3561574381906921507268638354899743020843286015762

199:

2.3586731583485945994132021040566845833150888010219

200:

2.3611762841937905728675685828778736919023860403764

Definition

For $n\geq1$, let $\mu_n=m(1+x_1+\cdots+x_{n-1})$, where $m$ is the logarithmic Mahler measure [5]. Equivalently, $\mu_n=W_n'(0)$, where $W_n(s)$ is the $s$-th moment of the distance from the origin after an $n$-step uniform random walk in the plane [1].

Parameters

$n$

— number of steps ($n\geq1$)

Formulas

(1)

For a polynomial $P$ in $k$ variables, $m(P)=\int_{[0,1]^k}\log|P(e^{2\pi i t_1},\dots,e^{2\pi i t_k})|\,dt_1\cdots dt_k$.

(2)

$W_n(s)=\int_{[0,1]^n}\left|\sum_{j=1}^n e^{2\pi i t_j}\right|^s dt_1\cdots dt_n =\int_{[0,1]^{n-1}}\left|1+\sum_{j=1}^{n-1}e^{2\pi i t_j}\right|^s dt_1\cdots dt_{n-1}$, and differentiating at $s=0$ gives $\mu_n=W_n'(0)$ [1].

(3)

For $n\geq3$, $\mu_n=\log2-\gamma-\int_0^1\frac{J_0(x)^n-1}{x}\,dx -\int_1^\infty\frac{J_0(x)^n}{x}\,dx$, where $J_0$ is the Bessel function of the first kind and $\gamma$ is Euler's constant [1].

(4)

$\mu_3=\frac{1}{\pi}\mathrm{Cl}_2(\pi/3) =\frac{3\sqrt3}{4\pi}L(2,\chi_{-3})$, where $\mathrm{Cl}_2(\pi/3)$ is in the table of Clausen-function values and $L(2,\chi_{-3})$ is in the table of Dirichlet $L$-values [2].

(5)

$\mu_4=7\zeta(3)/(2\pi^2)$, where $\zeta(3)$ is in the table of Riemann zeta values [2].

(6)

$\mu_5\stackrel{?}{=}\left(\frac{15}{4\pi^2}\right)^{5/2} \int_0^\infty\left(\eta(e^{-3t})^3\eta(e^{-5t})^3+ \eta(e^{-t})^3\eta(e^{-15t})^3\right)t^3\,dt$, with $\eta(q)=q^{1/24}\prod_{m\geq1}(1-q^m)$ [1].

(7)

$\mu_6\stackrel{?}{=}\left(\frac3{\pi^2}\right)^3 \int_0^\infty \eta(e^{-t})^2\eta(e^{-2t})^2 \eta(e^{-3t})^2\eta(e^{-6t})^2 t^4\,dt$ [1].

Comments

(8)

The index $n$ is the number of steps of the walk and the number of terms in $1+x_1+\cdots+x_{n-1}$. Multiplication by a monomial of modulus $1$ on the torus does not change $m$, so $m(x_1+\cdots+x_n)=m(1+x_1+\cdots+x_{n-1})$ and one variable can be removed. The values $\mu_1$ and $\mu_2$ are exactly zero, since $m(1)=m(1+x)=0$, and are not listed as rows.

(9)

Smyth evaluated $\mu_3=m(1+x+y)=\mathrm{Cl}_2(\pi/3)/\pi$ and $\mu_4=m(1+x+y+z)=7\zeta(3)/(2\pi^2)$ [2] [3]. Rodriguez Villegas conjectured the modular eta-integral evaluations for $\mu_5$ and $\mu_6$ quoted in (6) and (7); Borwein, Straub, Wan and Zudilin report numerical confirmations to 600 and 80 decimal places respectively [1], and Straub and Zudilin review the later short-walk literature [4].

(10)

The values approach $\tfrac12(\log n-\gamma)$ from above: the entries here give $n\bigl(\mu_n-\tfrac12(\log n-\gamma)\bigr)=0.12508667\ldots$ at $n=200$, tending to $1/8$, and the next correction is about $0.01736/n^2$. At $n=200$ the two terms $\tfrac12(\log n-\gamma)+\tfrac{1}{8n}$ agree with $\mu_{200}$ to six decimal digits only, so the fifty digits stored for each $n$ are not predicted by the asymptotics. The table stops at $n=200$ by choice and not by cost: $J_0(x)^n$ decays like $x^{-n/2}$, so a value from (3) takes less work as $n$ grows.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 80
mu3 = mp.clsin(2, mp.pi/3) / mp.pi
mu4 = 7 * mp.zeta(3) / (2 * mp.pi**2)

References

[1]

Jonathan M. Borwein, Armin Straub, James Wan and Wadim Zudilin, Densities of short uniform random walks, Canadian Journal of Mathematics 64 (2012), 961-990. (arXiv) (doi)

[2]

C. J. Smyth, On measures of polynomials in several variables, Bulletin of the Australian Mathematical Society 23 (1981), 49-63.

[3]

David W. Boyd, Speculations concerning the range of Mahler's measure, Canadian Mathematical Bulletin 24 (1981), 453-469. (doi)

[4]

Armin Straub and Wadim Zudilin, Short walk adventures, in From Analysis to Visualization: A Celebration of the Life and Legacy of Jonathan M. Borwein, Springer Proceedings in Mathematics & Statistics 313 (2020), 423-439. (arXiv)

Links

[5]

Wikipedia: Mahler measure

Similar tables

Pólya's random walk constants — gives return probabilities for simple random walks on $\mathbb Z^d$, indexed by the lattice dimension; the walks here are planar and indexed by their number of steps

Entropy constants of lattice models — holds $\mu_3/2$ as the honeycomb-lattice dimer entropy and $5\mu_3$ as the triangular-lattice spanning-tree entropy

Data properties

Entries are of type: real number

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

Table is complete: no (it holds $\mu_n$ from $n=3$ on: $n=3,4$ from closed forms, $n=5,6$ from conjectural eta-integral evaluations, and from $n=7$ on, as far as the table goes, from the Bessel integral (3); $\mu_1=\mu_2=0$)

How they were obtained:

The generator computes $\mu_3$ from the closed form $\mathrm{Cl}_2(\pi/3)/\pi$ and $\mu_4$ from $7\zeta(3)/(2\pi^2)$. It computes $\mu_5$ and $\mu_6$ from the Rodriguez Villegas eta-integrals (6) and (7), using the modular transformation of Dedekind's eta function for the small-$t$ part of the integral.

The eta-integral computation was repeated at 50, 80 and 120 working digits, and the 50 digits kept here agree. The identities equating those eta-integrals with $\mu_5$ and $\mu_6$ are conjectural; Borwein, Straub, Wan and Zudilin report numerical confirmations to 600 and 80 decimal places respectively [1].

From $n=7$ on, each value is computed from (3), taken from [1]. The integral over $[0,1]$ is smooth. The one over $[1,\infty)$ oscillates, and is taken between consecutive zeros of $J_0$ and summed with Richardson and Shanks acceleration: the zeros approach spacing $\pi$, and an oscillatory quadrature that assumes a fixed period stalls at about sixteen digits. Each value is computed at two working precisions, 55 and 63 digits, and kept to the digits both agree on, less two, and at most $50$. Before any of these values was published, the same computation reproduced the closed forms for $\mu_3$ and $\mu_4$ to all $50$ of their digits. No rigorous bound is carried for the tail of the integral, so these digits are agreement-checked rather than proven.