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Higher Mahler measure of cyclotomic polynomials

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Numbers
$n$
$k$ 
$m_k(\Phi_n)$
1
2:
0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527
comment: $\Phi_1(x)=x-1$; the moment is $\pi^2/12$.
1
3:
-1.803085354739391428099607242267174986147479438510748322688407333012757308679469635279683810414002887
comment: $\Phi_1(x)=x-1$; the moment is $-3\zeta(3)/2$.
3
2:
1.096622711232150957648276777764016792812633267471198958490372152913338313602133915889085933746505804
3
3:
-1.881167614185853462308291139495596441366927992937019922118602096539469053808212881057869125305520178
5
2:
1.315947253478581149177932133316820151375159920965438750188446583496005976322560699066903120495806964
5
3:
-1.636661662059463025106561334235254201220583435750330854950094081904587602416547223555591783253590443
7
2:
1.409943485869908374119212999982307305044814201034398660916192768031434974631315034714539057674078890
7
3:
-1.407000564826077670207099186423549518713147161565369650860785473981062123127439511938717454666100295
11
2:
1.495394606225660396793104696950931990199045364733453125214143844881824973093818976212389909654326096
11
3:
-1.098430747853158470858423983113297943804237294769499867204477699934140574416431849126810674201788362
13
2:
1.518400677090670556743767846134792482355953754960121634832822980956929972679877729692580523649008036
13
3:
-0.9929039525944422496493582794292806183428128183697345396195556852921391654307807163271120661549565987
15
2:
1.754596337971441532237242844422426868500213227953918333584595444661341301763414265422537493994409286
15
3:
-2.074855006906156013604465299651846058736303311353823345806231006222849692611915294016580960238021935
17
2:
1.548173239386566057856390745078611942794305789371104411986407745289418795673600822431650729995067017
17
3:
-0.8371909008932983895321905326888102245818190308350117945207955133259695145589398083985417839123375260
19
2:
1.558358589645688202973866999980444916102163064301177467328423585718954445645137669947648432166087194
19
3:
-0.7779636026008816249269326112425753696005935153149224168799126731391481404460734130795590037878638762
21
2:
1.879924647826544498825617333309743073393085601379198214554923690708579966175086712952718743565438520
21
3:
-1.865654750484632734892695863056562684136593217941417965763825147656535694668736345276180908246002386
23
2:
1.573415194376564417495353637661415398383343383763024592616620915049572362994366053232166774505856153
23
3:
-0.6836680949795585612202285192415829283282051264596204812806980530182232284486463433558459744357836177
29
2:
1.588212202474149662800952574692713975797606801165184698503297600771041695561711188529021007494939440
29
3:
-0.5816707242319395611672470887348338446064175596901604612763422682278851996153410948372473386998483777
31
2:
1.591871677595057841747498548367121150857048291490450101034411189712910455228904071451898936083637457
31
3:
-0.5547612874587382557978629395098515769550954658024168567128253983929515923807398449636169929261040206
33
2:
1.993859474967547195724139595934575986932060486311270833618858459842433297458425301616519879539101461
33
3:
-1.431238814859012157755954816222284301663908569666462190240871638403699093111694779758320467862917616
35
2:
2.255909577391853398590740799971691688071702721655037857465908428850295959410104055543262492278526224
35
3:
-1.953836807082720869222422271416866913824112073645634911088980146024236701406254863682180131567558205
37
2:
1.600476389365841938189376918898835319240059363336344425904867466414061322554465715081368660062467929
37
3:
-0.4884000502279594311112499103180318590030203044151762967259171877478563624581495537070628133054260805
39
2:
2.024534236120894075658357128179723309807938339946828846443763974609239963573170306256774031532010714
39
3:
-1.312881977400042672252746131209662554906780957879361531643562331828326046792136743722007717296802447
41
2:
1.604813723754367255095039186971731891920926732884681402668837296946348751612878901301101366458301176
41
3:
-0.4531077191105627296349729582380343637846565241923382581521969903532906277382386966877366746246635359
43
2:
1.606679786223849077484684581375187394120834787225244985695196410082332878068242713977032879675113154
43
3:
-0.4374992118168982204668624461004421248558506311364740656993416389372648721389057780584183193389213268
47
2:
1.609935469681242895270874418419514014980248839478994215656078267042986034862707238220147434649125541
47
3:
-0.4096048554137358143274709804736244708719890117064029958871406396489368542446756358635974816956705541
51
2:
2.064230985848754743808520993438149257059074385828139215981876993719225060898134429908867639993422689
51
3:
-1.097448215016193624328610404273161356075981027857944662941992203287744696679525535687423675923799932
53
2:
1.613897575020901409369162050294213393195950846467047523816019394853592235112574442251862317589197220
53
3:
-0.3744215619003084558034958620525989479428717064822774777625004111916594045599588222363714181910757620
55
2:
2.392631369961056634868967515121491184318472583573525000342630151810919956950110361939823855446921753
55
3:
-1.801231036467542828259973441949667553851238709163671693488793302702034893802657480818400381364912662
57
2:
2.077811452860917603965155999973926554802884085734903289771231447625272594193516893263531242888116259
57
3:
-1.028256676714248767436691334769609523014361899517278642003633469936962058427577088577655207684612252
59
2:
1.617053828427070056193221689245245101266086343559225582858684361075600564125180520039838580270271270
59
3:
-0.3453152009219115991482575253540849296417844389023715117933016203350502861103437477830556089918752611
61
2:
1.617967934604812888333523114733795268084213017580457479739893340363941774167082826721602197330910202
61
3:
-0.3366903361928890011189774481422905467562944402864602126669867667167239499193558514554014647996026574
65
2:
2.429441083345072890790028553815667971769526007936194615732516769531087956287804367508128837838412857
65
3:
-1.414606507799601140189466245318303538960497229203664264847079746645431448159347774278428199191892502
67
2:
1.620382812119148429957901507442353171469413335517144729709654375200305866367332204074918021506030963
67
3:
-0.3134492529508947058982198762687778380162697009979821633276017259151966596310607522493447395964883483
69
2:
2.097886925835419223327138183548553864511124511684032790155494553399429817325821404309555699341141537
69
3:
-0.8986075764957604299723801155672705393831132582857767869569832768806604537113274073191296953587688467
71
2:
1.621765981399659866944634671341151595004598494147547755513930648674655252510198044624704549906804357
71
3:
-0.2998205942887005781207876339421421038106589041634346423934185248406868896498958463908479144549013014
73
2:
1.622400723466743882548135506828956351010471135436842294752879349515623806425074834466044943077022285
73
3:
-0.2934842705389069492604485827603824549440305321712517216954724757227474291267237893648598429764230160
77
2:
2.563533610672560680216750909058740554626934910971633928938532305511699953875118244935525559407416164
77
3:
-1.551618093696714115091277367133794277723074806823794630689122620278459899566820592469794960649196113
79
2:
1.624112116634957747403144088587214743785798636634560482827513188491906110018350229861051319599255431
79
3:
-0.2761284981631094142038715918490813739167593722431428245641982709459065530771543317200606689029907948
83
2:
1.625115584115115274587205345361133319469323998782620143305009335040248344253764718727199636756870046
83
3:
-0.2657578181999425822201515309815132204259368744409872045429771146134865755004650994005323597344646161
85
2:
2.477077183018505692570225192125779108470889262993767059178252392463070073077761315890641167992107227
85
3:
-1.215436384737696243338977803400382641830591633208886206571514228794709513272387428675834317018813110
87
2:
2.117616269965532883734603432923618634396809068220246264671063467694722260748948251372028009993252586
87
3:
-0.7657254544052073277295779790356306607990760642975140402494033238732270356076501884887138605467960189
89
2:
1.626451661602740746174972299605058614059186419170766994614933979601805139275075021318644306230772652
89
3:
-0.2517115989589699122838884934676582412007423197788663001186584551985751919316857025440030187602012703
91
2:
2.602972589298292382989316307659644255467349294217351373999125110211879953165504679472995183398299490
91
3:
-1.746326620271453932490890999251577772261545090952071220673825220248424254865817851436872151037766950
93
2:
2.122495570126743788996664731156161534476064388653933468045881586283880606971872095269198581444849942
93
3:
-0.7333382734588785767135338658818800279399734029419467384449464781992049554276792227276946047161934913
95
2:
2.493373743433101124758187199968711865763460902881883947725477737150327113032220271916237491465739511
95
3:
-1.237478903689803236399286233497722454708647985436595020446872900713770532787040763462792401113143830
97
2:
1.627975983684842658776823257711530084175455572328377835284676185768254816069147256577612107829864286
97
3:
-0.2353309162709022293376169658690414821059449758141852485450243096738976539091100995479620896176822012
101
2:
1.628647590938838055913282343213886325959356337828513304688671514227730168716040469142206832296790797
101
3:
-0.2279852890066939811759150644138193369451027364000543498804438716824897088403249178255246607013108320
103
2:
1.628963833189505791458119873765966692236241649738771268437154751414958854185694069233302406438984349
103
3:
-0.2244978557039589888567836317312599669105269816431852326843722565270896042095679911585649595390080438
105
2:
3.007879436522471198120987733295588917428936962206717143287877905133727945880138740724349989704701633
105
3:
-1.787062413221423427725059814890261143208702373446922263425864107042718263982139055988674579877500057
Definition
The $k$-higher Mahler measure of the cyclotomic polynomial $\Phi_n(x)$ is $m_k(\Phi_n)=\int_0^1(\log|\Phi_n(e^{2\pi i t})|)^k\,dt$, with the natural logarithm and total circle measure one [1].
Parameters
$n$
—   cyclotomic order ($n\geq1$)
$k$
—   logarithmic moment order ($k\geq2$)
Formulas
(1)
$m_2(\Phi_n)=\dfrac{\pi^2}{12}\dfrac{\varphi(n)2^{\omega(n)}}{n}$, where $\omega(n)$ is the number of distinct prime divisors of $n$, including $\omega(1)=0$ [1], Theorem 5.
(2)
$m_3(\Phi_n)=-\dfrac32d^2\zeta(3)-3d\sum_{1\leq i<j\leq d}C_3(2\pi(\alpha_j-\alpha_i))-3\pi\sum_{1\leq i<j\leq d}S_2(2\pi(\alpha_j-\alpha_i))\bigl(d(\alpha_j-\alpha_i)-(j-i)\bigr)$ [1], Theorem 6. Here $d$ and $\alpha_j$ are the degree and ordered root angles, and $\zeta(3)$ is Apéry's constant.
(3)
$m_k(\Phi_n)=m_k(\Phi_{r(n)})$ for $k\geq2$, where $r(n)=\prod_{p\mid n,\ p\text{ odd}}p$ and the empty product is $1$. For $q=\operatorname{rad}(n)$, $\Phi_n(x)=\Phi_q(x^{n/q})$; for odd $r$, $|\Phi_{2r}(x)|=|\Phi_r(-x)|$ on the unit circle.
(4)
$\displaystyle\int_0^1|e^{2\pi i t}-1|^s\,dt=\frac{\Gamma(1+s)}{\Gamma(1+s/2)^2}$ for real $s>-1$, with $\Gamma$ the Gamma function. Differentiating at $s=0$ gives $m_2(\Phi_1)=\pi^2/12$ and $m_3(\Phi_1)=-3\zeta(3)/2$ [2], Theorem 14 and section 4.2.
(5)
$\operatorname{Li}_s(e^{2\pi i a/b})=b^{-s}\sum_{r=1}^b e^{2\pi i ar/b}\zeta(s,r/b)$ for $s>1$ and integers $a,b$ with $b\geq1$, obtained by grouping the absolutely convergent series by residue class. Here $\zeta(s,u)$ is the Hurwitz zeta function. Taking real and imaginary parts at $s=3$ and $s=2$ gives $C_3(2\pi a/b)$ and $S_2(2\pi a/b)$.
Comments
(6)
These are raw signed logarithmic moments. The absolute value encloses the polynomial, not its logarithm; there is no factorial divisor. The logarithmic singularities at roots of unity are integrable.
(7)
The ordinary logarithmic Mahler measure $m_1(\Phi_n)$ is zero. The second and third moments retain information about the fluctuations of $\log|\Phi_n|$ [1].
(8)
Write $d=\varphi(n)$ for the polynomial degree, where $\varphi$ is Euler's totient function. List the primitive-root angles as $0\leq\alpha_1<\cdots<\alpha_d<1$, with $\alpha_j=a_j/n$ and the coprime residues $a_j$ in increasing order; for $n=1$ the list is $\alpha_1=0$. The symbol $n$ always denotes the cyclotomic order, whereas $d$ denotes degree.
(9)
The functions $C_3(\theta)=\operatorname{Re}\operatorname{Li}_3(e^{i\theta})$ and $S_2(\theta)=\operatorname{Im}\operatorname{Li}_2(e^{i\theta})$ are the odd and even Clausen functions. Their absolutely convergent series fix the convention on the unit circle. That table uses $t=\theta/\pi$, so a root-angle difference $\alpha_j-\alpha_i$ corresponds to $t=2(\alpha_j-\alpha_i)$ before period reduction and reflection.
(10)
Power substitution and rotation preserve these logarithmic moments. Consequently the odd squarefree representatives suffice to recover every positive cyclotomic order by (3). This removes equalities forced by those transformations; it does not assert that different representatives have different moments.
(11)
For a prime $p$, $\Phi_p(x)=1+x+\cdots+x^{p-1}$, so its higher measures also belong to the geometric-sum family. For composite lengths the geometric sum is a product of several distinct cyclotomic factors, and its higher moments include mixed terms [1].
Programs
(P1)
Sage
# Download https://numberdb.org/files/T681/generate.py?raw=1 as generate.py.
# Install numberdb with sage -pip install numberdb, then run in Sage.
from generate import moment
moment(15, 3, digits=100)
References
[1]
M. Lalín and K. Sinha, Higher Mahler measure for cyclotomic polynomials and Lehmer's question (2011), Definition 1 and Theorems 5–6. (arXiv)
[2]
N. Kurokawa, M. Lalín and H. Ochiai, Higher Mahler measures and zeta functions, Acta Arithmetica 135 (2008), 269–297. (arXiv) (doi)
Similar tables
Cyclotomic_polynomials —   The integer polynomials whose logarithmic moments are tabulated here.
Values_of_the_Clausen_functions_at_rational_multiples_of_pi —   The Clausen values entering the finite formula for the third moment.
Values_of_the_Riemann_zeta_function_at_rational_numbers —   The Riemann zeta values entering the third moment and the linear-polynomial controls.
Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers —   The Hurwitz zeta values entering the finite residue-class evaluation of the Clausen functions.
Values_of_the_Gamma_function_at_rational_numbers —   The Gamma values related to the moment-generating integral for the linear polynomial.
Mahler_measures_of_x_x-1_y_y-1_k —   An ordinary Mahler-measure family in two variables; the present table instead varies cyclotomic order and logarithmic moment order.
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it holds both $k=2$ and $k=3$ for every odd squarefree $1\leq n\leq105$; rotation and power substitution reduce all orders to such representatives, and the noncanonical bound reaches $105=3\cdot5\cdot7$, covering one-, two-, and three-prime structures as well as the linear case. These first nonzero even and odd moments admit finite formulas at 100 significant digits; higher moment orders are not included)
How they were obtained:

The finite formulas of Lalín–Sinha are evaluated in Arb real and complex ball arithmetic. Exact rational root angles and pair weights are used, with 128 extra working bits for cancellation. Each computed enclosure must retain at least 32 bits beyond the requested precision, and the stored decimal interval contains that enclosure.

more

The minimum retained relative accuracy was 414 bits, about 124 decimal digits, at 461 working bits. The attached numerical evidence records the precision of each enclosure. Second moments are checked by exact Möbius divisor sums and by the Bernoulli-polynomial root-pair formula. Every third moment is recomputed using finite Hurwitz-zeta sums in place of polylogarithms. Direct tanh-sinh integration of the Möbius log-sine expression independently checks both moments over the full selection; that quadrature is a numerical check, while the finite ball computations justify the error bounds. The linear case is controlled by the Gamma ratio, and the polynomial identities for the arithmetic reduction and prime geometric sums are checked exactly. All stored intervals are read back and compared with the independent enclosures.



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