{"Title": "Higher Mahler measure of cyclotomic polynomials", "Parameters": {"n": {"type": "Z", "display": "$n$", "title": "cyclotomic order", "constraints": "$n\\geq1$"}, "k": {"type": "Z", "display": "$k$", "title": "logarithmic moment order", "constraints": "$k\\geq2$"}}, "Definition": "The $k$-higher Mahler measure of the cyclotomic polynomial HREF{Cyclotomic_polynomials}[$\\Phi_n(x)$] 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 CITE{LS}.", "Comments": {"conventions": "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.", "mean": "The ordinary logarithmic Mahler measure $m_1(\\Phi_n)$ is zero. The second and third moments retain information about the fluctuations of $\\log|\\Phi_n|$ CITE{LS}.", "root-notation": "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.", "clausen-notation": "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 HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}[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.", "representatives": "Power substitution and rotation preserve these logarithmic moments. Consequently the odd squarefree representatives suffice to recover every positive cyclotomic order by CITE{reduction}. This removes equalities forced by those transformations; it does not assert that different representatives have different moments.", "prime-overlap": "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 CITE{LS}."}, "Tags": ["number theory", "integral", "special values", "period"], "References": {"LS": {"bib": "M. Lalín and K. Sinha, Higher Mahler measure for cyclotomic polynomials and Lehmer's question (2011), Definition 1 and Theorems 5–6.", "arxiv": "1106.1304"}, "KLO": {"bib": "N. Kurokawa, M. Lalín and H. Ochiai, Higher Mahler measures and zeta functions, Acta Arithmetica 135 (2008), 269–297.", "arxiv": "0908.0171", "doi": "10.4064/aa135-3-5"}}, "Links": {}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "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", "rigour details": "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. 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."}, "Display properties": {"number-header": "$m_k(\\Phi_n)$"}, "Numbers": {"1": {"2": {"number": "0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527", "comment": "$\\Phi_1(x)=x-1$; the moment is $\\pi^2/12$."}, "3": {"number": "-1.803085354739391428099607242267174986147479438510748322688407333012757308679469635279683810414002887", "comment": "$\\Phi_1(x)=x-1$; the moment is $-3\\zeta(3)/2$."}}, "3": {"2": "1.096622711232150957648276777764016792812633267471198958490372152913338313602133915889085933746505804", "3": "-1.881167614185853462308291139495596441366927992937019922118602096539469053808212881057869125305520178"}, "5": {"2": "1.315947253478581149177932133316820151375159920965438750188446583496005976322560699066903120495806964", "3": "-1.636661662059463025106561334235254201220583435750330854950094081904587602416547223555591783253590443"}, "7": {"2": "1.409943485869908374119212999982307305044814201034398660916192768031434974631315034714539057674078890", "3": "-1.407000564826077670207099186423549518713147161565369650860785473981062123127439511938717454666100295"}, "11": {"2": "1.495394606225660396793104696950931990199045364733453125214143844881824973093818976212389909654326096", "3": "-1.098430747853158470858423983113297943804237294769499867204477699934140574416431849126810674201788362"}, "13": {"2": "1.518400677090670556743767846134792482355953754960121634832822980956929972679877729692580523649008036", "3": "-0.9929039525944422496493582794292806183428128183697345396195556852921391654307807163271120661549565987"}, "15": {"2": "1.754596337971441532237242844422426868500213227953918333584595444661341301763414265422537493994409286", "3": "-2.074855006906156013604465299651846058736303311353823345806231006222849692611915294016580960238021935"}, "17": {"2": "1.548173239386566057856390745078611942794305789371104411986407745289418795673600822431650729995067017", "3": "-0.8371909008932983895321905326888102245818190308350117945207955133259695145589398083985417839123375260"}, "19": {"2": "1.558358589645688202973866999980444916102163064301177467328423585718954445645137669947648432166087194", "3": "-0.7779636026008816249269326112425753696005935153149224168799126731391481404460734130795590037878638762"}, "21": {"2": "1.879924647826544498825617333309743073393085601379198214554923690708579966175086712952718743565438520", "3": "-1.865654750484632734892695863056562684136593217941417965763825147656535694668736345276180908246002386"}, "23": {"2": "1.573415194376564417495353637661415398383343383763024592616620915049572362994366053232166774505856153", "3": "-0.6836680949795585612202285192415829283282051264596204812806980530182232284486463433558459744357836177"}, "29": {"2": "1.588212202474149662800952574692713975797606801165184698503297600771041695561711188529021007494939440", "3": "-0.5816707242319395611672470887348338446064175596901604612763422682278851996153410948372473386998483777"}, "31": {"2": "1.591871677595057841747498548367121150857048291490450101034411189712910455228904071451898936083637457", "3": "-0.5547612874587382557978629395098515769550954658024168567128253983929515923807398449636169929261040206"}, "33": {"2": "1.993859474967547195724139595934575986932060486311270833618858459842433297458425301616519879539101461", "3": "-1.431238814859012157755954816222284301663908569666462190240871638403699093111694779758320467862917616"}, "35": {"2": "2.255909577391853398590740799971691688071702721655037857465908428850295959410104055543262492278526224", "3": "-1.953836807082720869222422271416866913824112073645634911088980146024236701406254863682180131567558205"}, "37": {"2": "1.600476389365841938189376918898835319240059363336344425904867466414061322554465715081368660062467929", "3": "-0.4884000502279594311112499103180318590030203044151762967259171877478563624581495537070628133054260805"}, "39": {"2": "2.024534236120894075658357128179723309807938339946828846443763974609239963573170306256774031532010714", "3": "-1.312881977400042672252746131209662554906780957879361531643562331828326046792136743722007717296802447"}, "41": {"2": "1.604813723754367255095039186971731891920926732884681402668837296946348751612878901301101366458301176", "3": "-0.4531077191105627296349729582380343637846565241923382581521969903532906277382386966877366746246635359"}, "43": {"2": "1.606679786223849077484684581375187394120834787225244985695196410082332878068242713977032879675113154", "3": "-0.4374992118168982204668624461004421248558506311364740656993416389372648721389057780584183193389213268"}, "47": {"2": "1.609935469681242895270874418419514014980248839478994215656078267042986034862707238220147434649125541", "3": "-0.4096048554137358143274709804736244708719890117064029958871406396489368542446756358635974816956705541"}, "51": {"2": "2.064230985848754743808520993438149257059074385828139215981876993719225060898134429908867639993422689", "3": "-1.097448215016193624328610404273161356075981027857944662941992203287744696679525535687423675923799932"}, "53": {"2": "1.613897575020901409369162050294213393195950846467047523816019394853592235112574442251862317589197220", "3": "-0.3744215619003084558034958620525989479428717064822774777625004111916594045599588222363714181910757620"}, "55": {"2": "2.392631369961056634868967515121491184318472583573525000342630151810919956950110361939823855446921753", "3": "-1.801231036467542828259973441949667553851238709163671693488793302702034893802657480818400381364912662"}, "57": {"2": "2.077811452860917603965155999973926554802884085734903289771231447625272594193516893263531242888116259", "3": "-1.028256676714248767436691334769609523014361899517278642003633469936962058427577088577655207684612252"}, "59": {"2": "1.617053828427070056193221689245245101266086343559225582858684361075600564125180520039838580270271270", "3": "-0.3453152009219115991482575253540849296417844389023715117933016203350502861103437477830556089918752611"}, "61": {"2": "1.617967934604812888333523114733795268084213017580457479739893340363941774167082826721602197330910202", "3": "-0.3366903361928890011189774481422905467562944402864602126669867667167239499193558514554014647996026574"}, "65": {"2": "2.429441083345072890790028553815667971769526007936194615732516769531087956287804367508128837838412857", "3": "-1.414606507799601140189466245318303538960497229203664264847079746645431448159347774278428199191892502"}, "67": {"2": "1.620382812119148429957901507442353171469413335517144729709654375200305866367332204074918021506030963", "3": "-0.3134492529508947058982198762687778380162697009979821633276017259151966596310607522493447395964883483"}, "69": {"2": "2.097886925835419223327138183548553864511124511684032790155494553399429817325821404309555699341141537", "3": "-0.8986075764957604299723801155672705393831132582857767869569832768806604537113274073191296953587688467"}, "71": {"2": "1.621765981399659866944634671341151595004598494147547755513930648674655252510198044624704549906804357", "3": "-0.2998205942887005781207876339421421038106589041634346423934185248406868896498958463908479144549013014"}, "73": {"2": "1.622400723466743882548135506828956351010471135436842294752879349515623806425074834466044943077022285", "3": "-0.2934842705389069492604485827603824549440305321712517216954724757227474291267237893648598429764230160"}, "77": {"2": "2.563533610672560680216750909058740554626934910971633928938532305511699953875118244935525559407416164", "3": "-1.551618093696714115091277367133794277723074806823794630689122620278459899566820592469794960649196113"}, "79": {"2": "1.624112116634957747403144088587214743785798636634560482827513188491906110018350229861051319599255431", "3": "-0.2761284981631094142038715918490813739167593722431428245641982709459065530771543317200606689029907948"}, "83": {"2": "1.625115584115115274587205345361133319469323998782620143305009335040248344253764718727199636756870046", "3": "-0.2657578181999425822201515309815132204259368744409872045429771146134865755004650994005323597344646161"}, "85": {"2": "2.477077183018505692570225192125779108470889262993767059178252392463070073077761315890641167992107227", "3": "-1.215436384737696243338977803400382641830591633208886206571514228794709513272387428675834317018813110"}, "87": {"2": "2.117616269965532883734603432923618634396809068220246264671063467694722260748948251372028009993252586", "3": "-0.7657254544052073277295779790356306607990760642975140402494033238732270356076501884887138605467960189"}, "89": {"2": "1.626451661602740746174972299605058614059186419170766994614933979601805139275075021318644306230772652", "3": "-0.2517115989589699122838884934676582412007423197788663001186584551985751919316857025440030187602012703"}, "91": {"2": "2.602972589298292382989316307659644255467349294217351373999125110211879953165504679472995183398299490", "3": "-1.746326620271453932490890999251577772261545090952071220673825220248424254865817851436872151037766950"}, "93": {"2": "2.122495570126743788996664731156161534476064388653933468045881586283880606971872095269198581444849942", "3": "-0.7333382734588785767135338658818800279399734029419467384449464781992049554276792227276946047161934913"}, "95": {"2": "2.493373743433101124758187199968711865763460902881883947725477737150327113032220271916237491465739511", "3": "-1.237478903689803236399286233497722454708647985436595020446872900713770532787040763462792401113143830"}, "97": {"2": "1.627975983684842658776823257711530084175455572328377835284676185768254816069147256577612107829864286", "3": "-0.2353309162709022293376169658690414821059449758141852485450243096738976539091100995479620896176822012"}, "101": {"2": "1.628647590938838055913282343213886325959356337828513304688671514227730168716040469142206832296790797", "3": "-0.2279852890066939811759150644138193369451027364000543498804438716824897088403249178255246607013108320"}, "103": {"2": "1.628963833189505791458119873765966692236241649738771268437154751414958854185694069233302406438984349", "3": "-0.2244978557039589888567836317312599669105269816431852326843722565270896042095679911585649595390080438"}, "105": {"2": "3.007879436522471198120987733295588917428936962206717143287877905133727945880138740724349989704701633", "3": "-1.787062413221423427725059814890261143208702373446922263425864107042718263982139055988674579877500057"}}, "Formulas": {"second": "$m_2(\\Phi_n)=\\dfrac{\\pi^2}{12}\\dfrac{\\varphi(n)2^{\\omega(n)}}{n}$, where $\\omega(n)$ is the number of distinct prime divisors of $n$, including $\\omega(1)=0$ CITE{LS}, Theorem 5.", "third": "$m_3(\\Phi_n)=-\\dfrac32d^2\\zeta(3)-3d\\sum_{1\\leq i<j\\leq d}C_3(2\\pi(\\alpha_j-\\alpha_i))-3\\pi\\sum_{1\\leq i<j\\leq d}S_2(2\\pi(\\alpha_j-\\alpha_i))\\bigl(d(\\alpha_j-\\alpha_i)-(j-i)\\bigr)$ CITE{LS}, Theorem 6. Here $d$ and $\\alpha_j$ are the degree and ordered root angles, and $\\zeta(3)$ is HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers#3}[Apéry's constant].", "reduction": "$m_k(\\Phi_n)=m_k(\\Phi_{r(n)})$ 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.", "linear": "$\\displaystyle\\int_0^1|e^{2\\pi i t}-1|^s\\,dt=\\frac{\\Gamma(1+s)}{\\Gamma(1+s/2)^2}$ for real $s>-1$, with $\\Gamma$ the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function]. Differentiating at $s=0$ gives $m_2(\\Phi_1)=\\pi^2/12$ and $m_3(\\Phi_1)=-3\\zeta(3)/2$ CITE{KLO}, Theorem 14 and section 4.2.", "residue-sum": "$\\operatorname{Li}_s(e^{2\\pi i a/b})=b^{-s}\\sum_{r=1}^b e^{2\\pi i ar/b}\\zeta(s,r/b)$ 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 HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}[Hurwitz zeta function]. Taking real and imaginary parts at $s=3$ and $s=2$ gives $C_3(2\\pi a/b)$ and $S_2(2\\pi a/b)$."}, "Similar tables": [{"table": "HREF{Cyclotomic_polynomials}", "relation": "The integer polynomials whose logarithmic moments are tabulated here."}, {"table": "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}", "relation": "The Clausen values entering the finite formula for the third moment."}, {"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}", "relation": "The Riemann zeta values entering the third moment and the linear-polynomial controls."}, {"table": "HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}", "relation": "The Hurwitz zeta values entering the finite residue-class evaluation of the Clausen functions."}, {"table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}", "relation": "The Gamma values related to the moment-generating integral for the linear polynomial."}, {"table": "HREF{Mahler_measures_of_x_x-1_y_y-1_k}", "relation": "An ordinary Mahler-measure family in two variables; the present table instead varies cyclotomic order and logarithmic moment order."}], "Keywords": ["higher Mahler measure", "logarithmic moment", "cyclotomic", "Lehmer"], "Programs": {"program-sage": {"language": "Sage", "code": "# Download https://numberdb.org/files/T681/generate.py?raw=1 as generate.py.\n# Install numberdb with sage -pip install numberdb, then run in Sage.\nfrom generate import moment\nmoment(15, 3, digits=100)", "comment": "With the attached generate.py in the working directory, this computes the third logarithmic moment for cyclotomic order 15. The moment function implements moment orders 2 and 3; running generate.py verifies the complete default selection."}}}