NumberDB$^\beta$ Advanced search Tags Tables Help Login
Mahler measures of $x+x^{-1}+y+y^{-1}+k$
edit · history · discussion · files · short url · special values dynamical systems number theory
Numbers
$k$ 
$m_k$
0:
0
comment: Here $x+x^{-1}+y+y^{-1}=x^{-1}y^{-1}(x+y)(xy+1)$, so the Mahler measure is zero.
1:
0.2513304337132522313748725666693362946368603913242386756813941352410479985658685459886503189769562038
comment: Rogers and Zudilin proved $m_1=\frac{15}{4\pi^2}L(E_{15},2)$ for the conductor $15$ elliptic curve $E_{15}$ [3]; by the functional equation this is $L'(E_{15},0)$.
2:
0.5114240670535037222832744264759064461855252405847380586488654506050566440840423506050438715969962838
3:
0.7947124479795412534370472266030718508172607025835369155763187298184466591964660533352741166779177608
4:
1.166243616123275120553537825873579675456264615943349081044006276446990547521755446906507297212536236
comment: This is $4G/\pi$, the square-lattice spanning-tree entropy.
equals: Entropy_constants_of_lattice_models#spanning-tree,square,entropy
5:
1.507982602279513388249235400016017767821162347945432054088364811446287991395211275931901913861737223
6:
1.727311754014289776616795004708050883619885248827014166175197648905362630966220265482256618960406496
7:
1.900643195244181505880567827194373581458427979838884429216049910502201756072278059689151959680379268
8:
2.045696268214014889133097705903625784742100962338952234595461802420226576336169402420175486387985135
9:
2.171019686780810765696065627165601461057559251463715586919598489645747177185379535370080852423282695
10:
2.281611585539835177186291231386738330029461296122833781561773095247084659649306130741094840860595095
11:
2.380710876143058481135303018574383770479644777154288826578058193562351450277986308697694670374770574
12:
2.470559870851386908558613522434745173612632823187219724860828737546412231235267069878886333097147920
13:
2.552785310961056999130150533925743672459380534475183490857378802498799741304052714535376034725797029
14:
2.628609678255289530539130308127266863822485207504981480574643426158380146923752807012216779758387967
15:
2.698977431089820143383749154779954356852911253815449954825376451854333170551299083862994046254092224
16:
2.764634770845774545123598233362699241005464304566625432495335487651527984224554005875153508746518242
17:
2.826182318473193003068255356240629120170548974011643495110100879293406598580391970362943946788373844
18:
2.884111167004966501716533595282300950651938919285449651866324044931433998543627680866747104924461762
19:
2.938828298528212563066887170948072544808099002768364770685531431358817185231347626015952033140683815
20:
2.990674957323486888149376676174963471935391921353834746797301664573564097676834983809823881449903229
21:
3.039940219424649077202415298175727104567501110690757706287459079038581769428106144257020604975553353
22:
3.086871203046491732726722480590542733607797781002709458601494480487979170857513654148540995019548578
23:
3.131680878153361035882600288786642351104919949511941938842141914356630111275957191046843394007548967
24:
3.174554126871033468908654494093422076717207498178866398041837097524795637088313380090310108556648039
25:
3.215652507719648608734309060656397765692626759578621150417352268569666397894469803277402159345935232
26:
3.255118044663917987456069975026799287387265162871529423408559866816355337895402414364882706059152539
27:
3.293076272413227639127685629290803422827476041613651474888985488800608609598003197254287843582742587
28:
3.329638707443472253087332273552183772686314544950888988394265230629004773299927512830184001195447860
29:
3.364904870597599900666880782157998552692437453272445067236882607561066748193900558645627237886841651
30:
3.398963955934911042035040469177774805747994479589477665618621574219758851230248642155843983421510825
31:
3.431896217877755437659216763352906190738200786387843621879690657272925556072004266026251851682982450
32:
3.463774132079939024373794428506230631414412196647243016658486185410362617507781969425386465001858591
33:
3.494663373076454593325392533626070696482197319442956317622928375243464291718643279743635320725292462
34:
3.524623642475277097872434979553492622542589763910740505801866045306689236565240615081140957745229980
35:
3.553709374386760827412254981489008215463034043015320379801838596027249109937157686451848449152528364
36:
3.581970339366830851216735896292127195275307934734594896438863584577123770035490676393779284934437435
37:
3.609452163956329210155551333402041465685031706780666280372553591497384476541374303323305202850086534
38:
3.636196779626515474888057875572197549372371218872692507926123156814464172374821261117760482970621113
39:
3.662242812367562879759234362523081536626726057270028722568512222072149510560838868744343998178682284
40:
3.687625922118824659503863775339550395489593186970276651027682022292663516070793336599931721598677650
41:
3.712379099614368203661888485639952227767045302225591013856786449126895022925300005644625184746615442
42:
3.736532926912843276109352852531656494230830119154859905546551477247145719146325726174846439226993766
43:
3.760115806827473813502167801809221467622031094155890815238495056247461483635565789545013640702632057
44:
3.783154165616606599084811414467390485018322536274791730575458057497859531583781694197486014029950057
45:
3.805672632596852983039049681345368288806867391975988611410423971563862199708763144382043261271976793
46:
3.827694199767685250189939762183860418843709019720337332310255046121160188126789054552753658301184866
47:
3.849240364063646645086271915493127530609035905097259403729261961493758686767625387002042193919402058
48:
3.870331254458680888958310098993129912129787142158362088065783905338634175430576116255787222242859268
49:
3.890985745821151618120563712222444482247660052976981413042609895814585347606691480174152405759905242
50:
3.911221561145734071048621486934108544409117587728239467660421453346147904891907850400335476898247872
51:
3.931055363559807702282039431973513518350534001036185466684477282298080106268244790833703732117893258
52:
3.950502839309468091896847956137303072979810648379928454573093245555069187143584412971345687832159397
53:
3.969578772767529457459015849239737596598796927927155061356039310939512826845189489192316548984602657
54:
3.988297114367819535831675997679668469084891393196254743768858008910796802291304939888642185800347850
55:
4.006671042252539702880080698844618768030158206970723535598183173198055780030353423273792604592899728
56:
4.024713018319095495644937803856180373397938355080600664430230971426947582382623632431161334663425650
57:
4.042434839266824195088368332684107348786249459550158445125041009774535820737090109443258930055963547
58:
4.059847683170174139538233175361140863015139399266877065478017045164803597212660406546234272398837798
59:
4.076962152041238138020183337542509469984019038390266405553539002746643723305653055273785573884368891
60:
4.093788310789545448943758424388115695842713910702635893926017952615385099061959692137998135496517679
61:
4.110335722939370206211977499451287224066217433791649974481058555770705807483102327058951919506108613
62:
4.126613483423430476403727382214992011905023931961913772728591095815307320424780868916871946305562392
63:
4.142630248735819223703922368456733943208597649004261161806450358616203163948667983930633372257547296
64:
4.158394264695559635764596187016365472536418332774913963335959981434255634422263033944499285706557796
65:
4.173913392044666085725616053173058229251864814892181413373839635296352779685964858692815068421450224
66:
4.189195130080471854007427688751174490109183405237487654036487382955930701024409694394890082681474134
67:
4.204246638500792095845612358901651384877547706296585332217634005996999017645782657464150471384830156
68:
4.219074757621831308586236175154331975119726881429950073515700988061914461888010250574223782044191324
69:
4.233686027112282749920116059443798730324080777620805672554788001591878374150230395653538178810764598
70:
4.248086703372515737054276824206849305523491713236982733267404072624422451595994315511744563592045990
71:
4.262282775674859498389726490886673189337450937024983358830646012161281132146295999708961267083468477
72:
4.276279981169557969742887777949561397260110539823675187350624644636516897548279716282558432636216613
73:
4.290083818850806754161906272013593260273072623196451764713016076408434702885104960473103513676807527
74:
4.303699562568234492028577883716506791725731075863883555391331931852108381546086047662139231750693401
75:
4.317132273161120455301852583145517614308291519717097458544322663448465173006284615507587587466388656
76:
4.330386809785430746077484012990937725227623778850506776809676095611019427908514162941946661303146356
77:
4.343467840497304996266714932942282866165056633703690605123541649841385215273347156905000809061752480
78:
4.356379852150845156635322205565689755179394643056652330798278392255824865214386006671329436691986157
79:
4.369127159662870451611305250736035489779480755791437074009113536551433347410885484106501480729193321
80:
4.381713914692640290314427787556529119194024968209492932183887372761279039634513972633929636320910249
81:
4.394144113780350749528308164927692928302457133668877400253754642660864238108981833818851208944700978
82:
4.406421605984428371976453843257097378493221711534133241772768048055133791578038796142549893175635642
83:
4.418550100054231968586010777004961862878207000407931901053533183366044132021294091616020793466838314
84:
4.430533171171688876480802351252286642289575064965948901666302799715795212813518504123462832823054718
85:
4.442374267292601470788146142507347303288697474260064984200153711229031152225947003302732883573648652
86:
4.454076715115831571505990677851614890949951801354618404228585875398840992657807560192879158456218894
87:
4.465643725706277256986071103001548724133272248536736955041910244854995416566341038855886620398913505
88:
4.477078399795474182369181998727980936062653695060544413899086507321901309418276900722103495580136111
89:
4.488383732781760257609606279332968542438268299180207275808619447125198558867696215027081167262488014
90:
4.499562619450219340425282291862531326855546901568765390370421444154365919648512471501897407598345724
91:
4.510617858431049445068653701629270958837789709078924908011325155658538046556655229315937981794146206
92:
4.521552156413568727142366278481290601438497583501500726367258112953822966113159066319137478507433205
93:
4.532368132131764691596852706528162110159644485560908491634973735383841547915621706528090594988075067
94:
4.543068320136096649583519767803524207293960399192476764990173851413283041024951459473856206595909522
95:
4.553655174365167664469470067692563222009767321397096833594948235920901147948526288490847440017451218
96:
4.564131071529880454623471127393324719325224182177785242551517314260178181188178503891386338321902948
97:
4.574498314321773339468384070607993875212367544330560488431186414138989921904281921754232936678975272
98:
4.584759134456389593303589701634644749240294556542110164895492718291709753023048478365266677119782563
99:
4.594915695561759564934012231973390204213932829690056233020354959958465036434261326878081896310862845
100:
4.604970095921363387794084006018928897341690971871818151115789398246065084745488434981564832848630620
Definition
For a nonnegative integer $k$, $m_k$ is the logarithmic Mahler measure [4] of the Laurent polynomial $x+x^{-1}+y+y^{-1}+k$: $m_k=\int_0^1\int_0^1\log|k+2\cos(2\pi s)+2\cos(2\pi t)|\,ds\,dt$.
Parameters
$k$
—   constant term ($k\geq0$)
Formulas
(1)
Jensen's formula in the variable $y$ gives $m_k=\frac{1}{2\pi}\int_0^{2\pi} \operatorname{arcosh}_+\left(|k+2\cos\theta|/2\right)\,d\theta$, where $\operatorname{arcosh}_+(u)=\operatorname{arcosh}(u)$ for $u>1$ and is $0$ for $0\leq u\leq1$.
(2)
For $k>4$, $m_k=\log k-\sum_{n=1}^{\infty} \binom{2n}{n}^2/(2n k^{2n})$ [2].
(3)
$m_4=4G/\pi$, where $G$ is Catalan's constant [2]; this is the square-lattice spanning-tree entropy.
Comments
(4)
Only nonnegative $k$ are listed. Replacing $x$ and $y$ by $-x$ and $-y$ gives $m_{-k}=m_k$.
(5)
Boyd conjectured from numerical evidence that, for integer $k\ne0,4$, $m_k=r_kL'(E_k,0)$ with $r_k\in\mathbb{Q}$, where $E_k$ is the elliptic curve $x+x^{-1}+y+y^{-1}+k=0$ [1]. The case $k=1$ is a theorem of Rogers and Zudilin [3].
References
[1]
David W. Boyd, Mahler's measure and special values of L-functions, Experimental Mathematics 7 (1998), 37-82. (doi)
[2]
Anthony J. Guttmann and Mathew D. Rogers, Spanning tree generating functions and Mahler measures, Journal of Physics A 45 (2012), 494001. (arXiv) (doi)
[3]
Mathew Rogers and Wadim Zudilin, On the Mahler measure of $1+X+1/X+Y+1/Y$, International Mathematics Research Notices 2014 (2014), 2305-2326. (arXiv) (doi)
Links
Similar tables
Entropy constants of lattice models —   holds lattice-model entropy constants; the square-lattice spanning-tree entropy is $m_4$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every integer $k$ with $0\leq k\leq100$: the factorisation at $k=0$, Rogers and Zudilin's theorem at $k=1$, the square-lattice entropy at $k=4$, and the first hundred positive cases of Boyd's integer family; for $k>100$, (2) gives $m_k$ from a few terms)
How they were obtained:

The generator computes $m_k$ in ball arithmetic. For $k=1,2,3$ it uses the one-dimensional integral from (1), after the substitution $\sin(\theta/2)=\sqrt{k}\sin(t)/2$ removes the endpoint singularity. For $k=4$ it evaluates $4G/\pi$ using arb's Hurwitz zeta function for $G=L(2,\chi_{-4})$. For $k\geq5$ it sums the absolutely convergent series in (2), with the remaining tail bounded by a geometric majorant.

more

Every entry was also checked against a computation not used for that row: a coefficient power series for $k=1,2,3$, the transformed Jensen integral for $k=4$, and direct arb integration of (1) for every $5\leq k\leq100$.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.