{"Title": "Higher Mahler measures of $(x^n-1)/(x-1)$", "Parameters": {"n": {"type": "Z", "display": "$n$", "title": "number of summands", "constraints": "$n\\geq2$"}, "k": {"type": "Z", "display": "$k$", "title": "logarithmic moment order", "constraints": "$k\\geq2$"}}, "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 CITE{LS}.", "Tags": ["number theory", "integral", "special values", "period"], "References": {"LS": {"bib": "M. Lal\u00edn and K. Sinha, Higher Mahler measure for cyclotomic polynomials and Lehmer's question (2011), Definition 1, Theorem 6 and section 5.1.", "arxiv": "1106.1304"}, "KLO": {"bib": "N. Kurokawa, M. Lal\u00edn and H. Ochiai, Higher Mahler measures and zeta functions, Acta Arithmetica 135 (2008), 269\u2013297.", "arxiv": "0908.0171", "doi": "10.4064/aa135-3-5"}}, "Links": {}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "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", "rigour details": "The second moment uses its exact rational multiple of pi squared; the third uses the finite root-difference formula of Lal\u00edn\u2013Sinha, 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.\n\nEvery 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."}, "Display properties": {"number-header": "$m_k(S_n)$"}, "Numbers": {"2": {"2": {"number": "0.8224670334241132182362075833230125946094749506033992188677791146850037352016004369168144503098793527", "equals": "HREF{T681#1,2}", "comment": "$S_2(x)=1+x$; the moment is $\\pi^2/12$."}, "3": {"number": "-1.803085354739391428099607242267174986147479438510748322688407333012757308679469635279683810414002887", "equals": "HREF{T681#1,3}", "comment": "$S_2(x)=1+x$; the signed third moment is $-3\\zeta(3)/2$."}}, "3": {"2": {"number": "1.096622711232150957648276777764016792812633267471198958490372152913338313602133915889085933746505804", "equals": "HREF{T681#3,2}"}, "3": {"number": "-1.881167614185853462308291139495596441366927992937019922118602096539469053808212881057869125305520178", "equals": "HREF{T681#3,3}"}}, "4": {"2": "1.233700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029", "3": 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"3": "-0.3911357067917552618869017669098640732383696601609990636388721710499144941055169050957914293924260890"}}, "Comments": {"conventions": "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 $S_n(1)=n$.", "cyclotomic": "The polynomial $S_n$ factors into HREF{Cyclotomic_polynomials}[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 HREF{T681}[cyclotomic higher measures]. For composite $n$, the powers of the sum of factor logarithms contain mixed terms.", "clausen": "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 HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}[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.", "dirichlet": "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 HREF{Lebesgue_constants_of_the_Fourier_partial_sums}[Dirichlet kernel]. Its magnitude therefore has the same logarithmic moments. The Fourier Lebesgue constants integrate $|D_q|$ itself.", "phases": "The phases $1,x,\\ldots,x^{n-1}$ depend on one circle variable. The HREF{Mahler_measures_of_(short_random_walks)}[ordinary Mahler measures of sums of independent phases] and HREF{Moments_of_uniform_random_walks_in_the_plane}[radial moments of planar random walks] integrate independently varying phases and give different quantities."}, "Formulas": {"sine": "$m_k(S_n)=\\int_0^1\\left(\\log\\left|\\dfrac{\\sin(n\\pi t)}{\\sin(\\pi t)}\\right|\\right)^k\\,dt$. The values at the isolated zeros are understood through this convergent improper integral.", "second": "$m_2(S_n)=\\dfrac{\\pi^2}{6}\\left(1-\\dfrac1n\\right)$, obtained by Parseval from the Fourier series for $\\log|2\\sin(\\pi t)|$.", "root-pairs": "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 CITE{LS}, Theorem 6. Each root-angle difference $r/n$ occurs $d-r$ times; empty sums are zero. Here $\\zeta(3)$ is HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers#3}[Ap\u00e9ry's constant].", "hurwitz": "$m_3(S_n)=\\dfrac{9n-6-3n^3}{2n^2}\\zeta(3)+\\dfrac{3\\pi}{2n^2}\\sum_{r=1}^{n-1}\\cot(\\pi r/n)\\zeta(2,r/n)$. This follows by grouping the absolutely convergent cotangent series of CITE{LS}, section 5.1, item 1, by residue class modulo $n$. The function $\\zeta(s,a)$ is the HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}[Hurwitz zeta function].", "linear": "$\\displaystyle\\int_0^1|S_2(e^{2\\pi i t})|^s\\,dt=\\dfrac{\\Gamma(1+s)}{\\Gamma(1+s/2)^2}$ for real $s>-1$, where $\\Gamma$ is the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function]. Differentiating at $s=0$ gives $m_2(S_2)=\\pi^2/12$ and $m_3(S_2)=-3\\zeta(3)/2$ CITE{KLO}, Theorem 14 and section 4.2."}, "Similar tables": [{"table": "HREF{T681}", "relation": "The same logarithmic moments for individual cyclotomic factors; prime geometric sums coincide with those factors."}, {"table": "HREF{Cyclotomic_polynomials}", "relation": "The polynomial factors of the geometric sums."}, {"table": "HREF{Values_of_the_Clausen_functions_at_rational_multiples_of_pi}", "relation": "The Clausen values used in the root-difference formula for the third moment."}, {"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}", "relation": "The Riemann zeta values appearing in the finite third-moment formulas."}, {"table": "HREF{Values_of_the_Hurwitz_zeta_function_at_pairs_of_rational_numbers}", "relation": "The Hurwitz zeta values used in the cotangent formula for the third moment."}, {"table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}", "relation": "The Gamma function in the moment-generating integral for the length-two case."}, {"table": "HREF{Lebesgue_constants_of_the_Fourier_partial_sums}", "relation": "The mean absolute Dirichlet kernel; odd-length geometric sums have the same magnitude but this table integrates powers of its logarithm."}, {"table": "HREF{Mahler_measures_of_(short_random_walks)}", "relation": "Ordinary logarithmic measures of sums whose phases vary independently."}, {"table": "HREF{Moments_of_uniform_random_walks_in_the_plane}", "relation": "Radial power moments of sums of independent unit phases."}], "Keywords": ["higher Mahler measure", "logarithmic moment", "geometric sum", "cyclotomic", "Lehmer", "Dirichlet kernel"], "Programs": {"program-sage": {"code": "# After downloading generate.py from this table:\nfrom generate import moment\nmoment(6, 3, digits=100)", "comment": "The attached generator evaluates either supported moment for any integer length n >= 2, using certified ball arithmetic.", "language": "Sage"}}}