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Higher Mahler measures of $(x^n-1)/(x-1)$

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Numbers
$n$
$k$ 
$m_k(S_n)$
2
2:
0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527
equals: T681#1,2
comment: $S_2(x)=1+x$; the moment is $\pi^2/12$.
2
3:
-1.803085354739391428099607242267174986147479438510748322688407333012757308679469635279683810414002887
equals: T681#1,3
comment: $S_2(x)=1+x$; the signed third moment is $-3\zeta(3)/2$.
3
2:
1.096622711232150957648276777764016792812633267471198958490372152913338313602133915889085933746505804
equals: T681#3,2
3
3:
-1.881167614185853462308291139495596441366927992937019922118602096539469053808212881057869125305520178
equals: T681#3,3
4
2:
1.233700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029
4
3:
-1.769026899833204342169804192869393878075461504894756975430081045685763757705523605313148387300718748
5
2:
1.315947253478581149177932133316820151375159920965438750188446583496005976322560699066903120495806964
equals: T681#5,2
5
3:
-1.636661662059463025106561334235254201220583435750330854950094081904587602416547223555591783253590443
equals: T681#5,3
6
2:
1.370778389040188697060345972205020991015791584338998698112965191141672892002667394861357417183132254
6
3:
-1.514385940945442422234473750425135811642780510045585279513050609448685186301286073299197133175404421
7
2:
1.409943485869908374119212999982307305044814201034398660916192768031434974631315034714539057674078890
equals: T681#7,2
7
3:
-1.407000564826077670207099186423549518713147161565369650860785473981062123127439511938717454666100295
equals: T681#7,3
8
2:
1.439317308492198131913363270815272040566581163555948633018613450698756536602800764604425288042288867
8
3:
-1.313655608403104359602386870931669826323378046957637012171598006058223308793829617751619765557457470
9
2:
1.462163614976201276864369037018689057083511023294931944653829537217784418136178554518781244995341071
9
3:
-1.232373984651885283218556813635987873334539074279962661932716367944368882780177574210888229854567703
10
2:
1.480440660163403792825173649981422670297054911086118593962002406433006723362880786450266010557782835
10
3:
-1.161193253936568043540087632849240778990216322731973979231465280612479941709423637680772782323277976
11
2:
1.495394606225660396793104696950931990199045364733453125214143844881824973093818976212389909654326096
equals: T681#11,2
11
3:
-1.098430747853158470858423983113297943804237294769499867204477699934140574416431849126810674201788362
equals: T681#11,3
12
2:
1.507856227944207566766380569425523090117370742772898567924261710255840181202934134347493158901445480
12
3:
-1.042706568731004557029877747518168904352455221133622088144166845077544715705435362224332512809545342
13
2:
1.518400677090670556743767846134792482355953754960121634832822980956929972679877729692580523649008036
equals: T681#13,2
13
3:
-0.9929039525944422496493582794292806183428128183697345396195556852921391654307807163271120661549565987
equals: T681#13,3
14
2:
1.527438776359067405295814083314166247131882051120598549325875498700721222517257954274083979146918798
14
3:
-0.9481199925551626095988624919113914029554199136697366439688903695850726448635908327334495854529286228
15
2:
1.535271795725011340707587488869623509937686574459678541886521014078673639042987482244720307245108125
15
3:
-0.9076215895756826324765710652681868974064619675890703310397543625590260823874749741877167318040633698
16
2:
1.542125687670212284192889218730648614892765532381373535377085840034382003503000819219027094331023786
16
3:
-0.8708096369301008621643839145922703147947242731487459782683873334510199764628673993467779188622403773
17
2:
1.548173239386566057856390745078611942794305789371104411986407745289418795673600822431650729995067017
equals: T681#17,2
17
3:
-0.8371909008932983895321905326888102245818190308350117945207955133259695145589398083985417839123375260
equals: T681#17,3
18
2:
1.553548840912213856668392101832357123151230462250865191194693883293895944269689714176205072807549888
18
3:
-0.8063562144061197299071346765572768242913537525575619098379364349059937284154467214153822429978171259
19
2:
1.558358589645688202973866999980444916102163064301177467328423585718954445645137669947648432166087194
equals: T681#19,2
19
3:
-0.7779636026008816249269326112425753696005935153149224168799126731391481404460734130795590037878638762
equals: T681#19,3
20
2:
1.562687363505815114648794408313723929758002406146458515848780317901507096883040830141947455588770770
20
3:
-0.7517251877301875851954610962410698391417245551383393238487928255236388045125508988049482227297939325
21
2:
1.566603873188787082354681111091452561160904667815998512129103075590483305145905594127265619637865434
21
3:
-0.7273969726701548365741256462514955379694921911780537312094931537282011305939624980502324471488089253
22
2:
1.570164336536943416632759931798478589708997632970125781474851037125916221748509925023009405137042401
22
3:
-0.7047708177315172552052242170350860573908012498151135002462377067288597732868608380025467882702687990
23
2:
1.573415194376564417495353637661415398383343383763024592616620915049572362994366053232166774505856153
equals: T681#23,2
23
3:
-0.6836680949795585612202285192415829283282051264596204812806980530182232284486463433558459744357836177
equals: T681#23,3
24
2:
1.576395147396217001619397868035774139668160321989848502829909969812923825803067504090561029760602093
24
3:
-0.6639346327405529109488427668482213494579098484417474114789154531996307614769579333192073460677380083
25
2:
1.579136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
25
3:
-0.6454366588672624372202748034081387103647998395905792536818374000537419846384945578028130844148833641
26
2:
1.581667371969448496608091506390408835787451828083460036284190605163468721541539301763104712134383370
26
3:
-0.6280575225245866443583858836349183358804958303155052431380799922222957257845323512428667107904980226
27
2:
1.584010582890884716603066456770246478507136941902842940041648665319266452980860100728679682078286161
27
3:
-0.6116950271039239243490478066791329798057500683297472994725349819128397192788328311950447752225476835
28
2:
1.586186421603646920884114624980095718175415976163698493530716864035364346460229414053856439883338752
28
3:
-0.5962592462258359618423698886417711421058511888774987914489477467204938062656289508136035410892765264
29
2:
1.588212202474149662800952574692713975797606801165184698503297600771041695561711188529021007494939440
equals: T681#29,2
29
3:
-0.5816707242319395611672470887348338446064175596901604612763422682278851996153410948372473386998483777
equals: T681#29,3
30
2:
1.590102931286618888590001327757824349578318237833238489811039621724340554723094178039174603932433415
30
3:
-0.5678589847209303087722209748362546270182449863215237567491874525780373541838581681143279433681837870
31
2:
1.591871677595057841747498548367121150857048291490450101034411189712910455228904071451898936083637457
equals: T681#31,2
31
3:
-0.5547612874587382557978629395098515769550954658024168567128253983929515923807398449636169929261040206
equals: T681#31,3
32
2:
1.593529877259219360332652192688336902055857716794085986556322034702194736953100846526327997475391246
32
3:
-0.5423215867778309488726545687541605253075112785297470667660137737680580929223022227698543850901066070
33
2:
1.595087579974037756579311676747660789545648389049016666895086767873946637966740241293215903631281169
33
3:
-0.5304896543888438673630102611910348268574984362126963609164485685344419044031267283357964206489078939
34
2:
1.596553653117396247164402955862318566006627845288951424860982987329713133038400848132639815307412861
34
3:
-0.5192203371013334938789750123524201882160506364177749281541910661036030855714706527874163122432072191
35
2:
1.597935950652562824001774733313281612384122761172318482371685137102292971248823706009810932030622742
35
3:
-0.5084729258361167054272058704930166753777564253453890424880871192736027205224608579511926878039482047
36
2:
1.599241453880220146570403634239191156185090181728831814465126056331951707336445294004916986713654297
36
3:
-0.4982106169141686102527558128317998154345064435415368061455999678407762981502509075828128971344600792
37
2:
1.600476389365841938189376918898835319240059363336344425904867466414061322554465715081368660062467929
equals: T681#37,2
37
3:
-0.4884000502279594311112499103180318590030203044151762967259171877478563624581495537070628133054260805
equals: T681#37,3
38
2:
1.601646328246957319723141083313235052660556482753987952531990907544480958024169271890638666392922950
38
3:
-0.4790109117664529876340496807462479426539847141735321148563266475870538173670219116955361815423180236
39
2:
1.602756270262374476562866059808947620264617852457906170101313146565648304495426492453279441629508482
39
3:
-0.4700155902452287122199031829368019388283210833622890522304599457795356004487139817371609745284654410
40
2:
1.603810715177020775560604787479874559488476153676628476792169273635757283643120851987788178104264738
40
3:
-0.4613888794176654219214023443741542485972067732227747045068245877879064863837164331692127397066144906
41
2:
1.604813723754367255095039186971731891920926732884681402668837296946348751612878901301101366458301176
equals: T681#41,2
41
3:
-0.4531077191105627296349729582380343637846565241923382581521969903532906277382386966877366746246635359
equals: T681#41,3
42
2:
1.605768970018506759413548138868738875189927284511398474932330652480245387774553233980447260128812069
42
3:
-0.4451509692137727627613283200234488245887811107010539443019238293510911734999945617929783541889356363
43
2:
1.606679786223849077484684581375187394120834787225244985695196410082332878068242713977032879675113154
equals: T681#43,2
43
3:
-0.4374992118168982204668624461004421248558506311364740656993416389372648721389057780584183193389213268
equals: T681#43,3
44
2:
1.607549201692584926552587549222251889463973767088462109605204633247961846075855399428319152878400553
44
3:
-0.4301345774723253777159138811875292867301318622631708497586635272811190913126391977163400773265481571
45
2:
1.608379976473821404550805940720557962791862125624425139119212490939562859949796409970659369494875179
45
3:
-0.4230405922082751194138963653311470439360881750269970403276271465188519169094723218959686135206908778
46
2:
1.609174630612395426983884402153720293801146642484911515176089572209789916698783463532897837562807429
46
3:
-0.4162020424460055079202049253847308940448051108951936708719354796482353192814945838489006190963221606
47
2:
1.609935469681242895270874418419514014980248839478994215656078267042986034862707238220147434649125541
equals: T681#47,2
47
3:
-0.4096048554137358143274709804736244708719890117064029958871406396489368542446756358635974816956705541
equals: T681#47,3
48
2:
1.610664607122221719045906517340899664443555111598323470282734099591465648103134188962094965190180399
48
3:
-0.4032359930136783475678297350297357027270754651429271461390986511819788106007630584566189972100600397
49
2:
1.611363983851323856136243428551208348622644801182169898189934592035925685292931468245187494484661589
49
3:
-0.3970833574016089923406519747574303888919629068388987193820624274131081610679821299787438995895301202
50
2:
1.612035385511261907742966863313104685434570903182662468980847064782607320995136856356956322607363531
50
3:
-0.3911357067917552618869017669098640732383696601609990636388721710499144941055169050957914293924260890
Definition
For the geometric-sum polynomial $S_n(x)=1+x+\cdots+x^{n-1}=(x^n-1)/(x-1)$, its $k$-higher Mahler measure is $m_k(S_n)=\int_0^1(\log|S_n(e^{2\pi i t})|)^k\,dt$, with the natural logarithm and total circle measure one [1].
Parameters
$n$
—   number of summands ($n\geq2$)
$k$
—   logarithmic moment order ($k\geq2$)
Formulas
(1)
$m_k(S_n)=\int_0^1\left(\log\left|\dfrac{\sin(n\pi t)}{\sin(\pi t)}\right|\right)^k\,dt$. The values at the isolated zeros are understood through this convergent improper integral.
(2)
$m_2(S_n)=\dfrac{\pi^2}{6}\left(1-\dfrac1n\right)$, obtained by Parseval from the Fourier series for $\log|2\sin(\pi t)|$.
(3)
Let $d=n-1$ be the degree. Then $m_3(S_n)=-\dfrac32d^2\zeta(3)-3d\sum_{r=1}^{n-2}(d-r)C_3(2\pi r/n)+\dfrac{3\pi}{n}\sum_{r=1}^{n-2}r(d-r)S_2(2\pi r/n)$, by [1], Theorem 6. Each root-angle difference $r/n$ occurs $d-r$ times; empty sums are zero. Here $\zeta(3)$ is Apéry's constant.
(4)
$m_3(S_n)=\dfrac{9n-6-3n^3}{2n^2}\zeta(3)+\dfrac{3\pi}{2n^2}\sum_{r=1}^{n-1}\cot(\pi r/n)\zeta(2,r/n)$. This follows by grouping the absolutely convergent cotangent series of [1], section 5.1, item 1, by residue class modulo $n$. The function $\zeta(s,a)$ is the Hurwitz zeta function.
(5)
$\displaystyle\int_0^1|S_2(e^{2\pi i t})|^s\,dt=\dfrac{\Gamma(1+s)}{\Gamma(1+s/2)^2}$ for real $s>-1$, where $\Gamma$ is the Gamma function. Differentiating at $s=0$ gives $m_2(S_2)=\pi^2/12$ and $m_3(S_2)=-3\zeta(3)/2$ [2], Theorem 14 and section 4.2.
Comments
(6)
These are raw signed logarithmic moments. The absolute value encloses the polynomial, not its logarithm; there is no factorial divisor, exponentiation or standardization. The zeros at the nontrivial $n$th roots of unity give integrable logarithmic singularities. At $x=1$, the polynomial has the removable quotient value $S_n(1)=n$.
(7)
The polynomial $S_n$ factors into cyclotomic polynomials as $S_n(x)=\prod_{d\mid n,\ d>1}\Phi_d(x)$. Thus its ordinary logarithmic Mahler measure is zero. For a prime $p$, $S_p=\Phi_p$, so its higher measures equal the corresponding cyclotomic higher measures. For composite $n$, the powers of the sum of factor logarithms contain mixed terms.
(8)
Write $C_3(\theta)=\operatorname{Re}\operatorname{Li}_3(e^{i\theta})$ and $S_2(\theta)=\operatorname{Im}\operatorname{Li}_2(e^{i\theta})$, the odd and even Clausen functions. The companion uses $t=\theta/\pi$, so the root difference $r/n$ here corresponds to $t=2r/n$ before reduction by period and reflection.
(9)
For odd length $n=2q+1$, $S_n(e^{i\theta})=e^{iq\theta}D_q(\theta)$, where $D_q(\theta)=1+2\sum_{j=1}^q\cos(j\theta)$ is the Dirichlet kernel. Its magnitude therefore has the same logarithmic moments. The Fourier Lebesgue constants integrate $|D_q|$ itself.
(10)
The phases $1,x,\ldots,x^{n-1}$ depend on one circle variable. The ordinary Mahler measures of sums of independent phases and radial moments of planar random walks integrate independently varying phases and give different quantities.
Programs
(P1)
Sage
# SageMath with numberdb installed; no table attachment is needed.
# Return a certified real ball using 512-bit Arb arithmetic by default.
# Increase bits for a narrower enclosure; the ball displays its uncertainty.
# The attached generator reproduces the full table.
import numberdb.sage
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField


def moment(n, k, bits=512):
    n, k = ZZ(n), ZZ(k)
    if n < 2 or k not in (2, 3):
        raise ValueError('Require integer n >= 2 and k = 2 or 3.')
    R, C = RealBallField(bits), ComplexBallField(bits)
    pi = R.pi()
    if k == 2:
        return pi**2 * QQ(n - 1) / (6*n)
    total = R(0)
    for r in range(1, n):
        if 2*r == n:
            continue  # cot(pi/2) is exactly zero.
        a = QQ(r) / n
        theta = pi * R(a)
        total += theta.cos()/theta.sin() * C(2).zeta(C(a)).real()
    return (QQ(9*n - 6 - 3*n**3)/(2*n**2) * R(3).zeta()
            + 3*pi*total/(2*n**2))


moment(6, 3)
References
[1]
M. Lalín and K. Sinha, Higher Mahler measure for cyclotomic polynomials and Lehmer's question (2011), Definition 1, Theorem 6 and section 5.1. (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
T681 —   The same logarithmic moments for individual cyclotomic factors; prime geometric sums coincide with those factors.
Cyclotomic_polynomials —   The polynomial factors of the geometric sums.
Values_of_the_Clausen_functions_at_rational_multiples_of_pi —   The Clausen values used in the root-difference formula for the third moment.
Values_of_the_Riemann_zeta_function_at_rational_numbers —   The Riemann zeta values appearing in the finite third-moment formulas.
Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers —   The Hurwitz zeta values used in the cotangent formula for the third moment.
Values_of_the_Gamma_function_at_rational_numbers —   The Gamma function in the moment-generating integral for the length-two case.
Lebesgue_constants_of_the_Fourier_partial_sums —   The mean absolute Dirichlet kernel; odd-length geometric sums have the same magnitude but this table integrates powers of its logarithm.
Mahler_measures_of_(short_random_walks) —   Ordinary logarithmic measures of sums whose phases vary independently.
Moments_of_uniform_random_walks_in_the_plane —   Radial power moments of sums of independent unit phases.
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 every length $2\leq n\leq50$ at $k=2,3$, the first nonzero even and odd logarithmic moments. The noncanonical prefix includes primes, prime powers, and products of two and three distinct primes, with degree at most 49; these bounds permit arithmetic breadth and independent checks of the finite formulas at 100 significant digits. Higher moment orders are not included)
How they were obtained:

The second moment uses its exact rational multiple of pi squared; the third uses the finite root-difference formula of Lalín–Sinha, Theorem 6. All transcendental evaluations use Arb real and complex balls, including exact rational angles. Computation at 461 bits retained at least 418 bits of relative accuracy, about 125 decimal digits, and each stored 100-digit decimal interval contains the computed enclosure. The 128 extra working bits accommodate cancellation; the generator also checks the resulting radius.

more

Every second moment is checked independently with exact Bernoulli-polynomial root pairs. Every third moment is checked at 560 bits using the cotangent formula with Hurwitz zeta instead of polylogarithms, and the generator is repeated at 130 requested digits. Direct tanh-sinh integration of the log-sine quotient checks all 98 moments at 45 decimal working digits, with absolute discrepancies below 10^(-34). This integration is a numerical check, not the source of certification. The length-two controls follow from the Gamma ratio. Cyclotomic factorization and the odd-length Dirichlet-kernel identity are checked as exact polynomial identities. All 30 prime-case overlaps are compared with the stored cyclotomic companion. After submission, every stored interval is required to contain the independent enclosure. The attached files provide the algorithms and measured evidence.



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