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{
"Title": "Minimal polynomials of Bettin–Conrey cotangent sums",
"Definition": "For an integer $k\\geq2$, let $\\alpha_k=c_0(1/k)=-\\sum_{m=1}^{k-1}(m/k)\\cot(\\pi m/k)$ be the HREF{Bettin_Conrey_cotangent_sums}[Bettin–Conrey cotangent sum], with $\\pi$ positive CITE{BC}. The polynomial $P_k(x)$ is the unique irreducible polynomial in $\\mathbb Z[x]$ that vanishes at $\\alpha_k$, has coefficient gcd $1$, and has positive leading coefficient.",
"Parameters": {
"k": {
"type": "Z",
"display": "$k$",
"title": "Denominator",
"constraints": "$k\\geq2$"
}
},
"Comments": {
"normalization": "This primitive integer normalization is obtained by clearing denominators in the monic minimal polynomial over $\\mathbb Q$. It need not be monic: $\\alpha_4=1/2$ gives $P_4(x)=2x-1$. The number $\\alpha_k$ is not multiplied by $k$, $\\pi$ or $i$, and $x$ is the polynomial variable.",
"zero": "The omitted case $k=2$ has $\\alpha_2=0$ and $P_2(x)=x$.",
"field": "Writing $L=\\operatorname{lcm}(4,k)$ and $\\zeta_L=e^{2\\pi i/L}$ places $\\alpha_k$ in the maximal real subfield of $\\mathbb Q(\\zeta_L)$, so $\\deg P_k\\leq\\varphi(L)/2$, where $\\varphi$ is Euler’s totient function. The cotangent definition specifies the real embedding.",
"numerators": "For $1\\leq h<k$ with $\\gcd(h,k)=1$, the primitive minimal polynomial of $c_0(h/k)$ is either $P_k(x)$ or $(-1)^{\\deg P_k}P_k(-x)$, by CITE{formula-galois}. Thus numerator $1$ represents the reduced-numerator family up to conjugacy and a sign change."
},
"Formulas": {
"formula-cyclotomic": "With $L=\\operatorname{lcm}(4,k)$, $z=\\zeta_L$, $w=z^{L/k}$ and $i=z^{L/4}$, $\\displaystyle\\alpha_k=-\\frac{i}{k}\\sum_{m=1}^{k-1}m\\frac{w^m+1}{w^m-1}$. This follows from $\\cot\\theta=i(e^{2i\\theta}+1)/(e^{2i\\theta}-1)$.",
"formula-galois": "For $\\gcd(a,L)=1$, the automorphism $\\sigma_a(z)=z^a$ satisfies $\\sigma_a(\\alpha_k)=(-1)^{(a-1)/2}c_0(a/k)$, using the same finite-sum definition for any numerator coprime to $k$. Indeed, $\\sigma_a(i)=i^a$; every reduced residue modulo $k$ has a lift coprime to $L$.",
"formula-estermann": "$\\displaystyle D(0,1/k)=\\frac14+\\frac{i}{2}\\alpha_k=\\sum_{r=1}^k\\sum_{s=1}^k\\left(\\frac12-\\frac rk\\right)\\left(\\frac12-\\frac sk\\right)e^{2\\pi i rs/k}$. Here $D(s,h/k)=\\sum_{n\\geq1}d(n)e^{2\\pi i nh/k}n^{-s}$ for $\\Re s>1$, $d(n)$ counts positive divisors, and $D(0,h/k)$ is its analytic continuation CITE{BC}, equation (12). The finite expression follows by grouping both divisor variables modulo $k$ and using $\\zeta(0,u)=1/2-u$ CITE{DLMFZero}.",
"formula-vasyunin": "$\\displaystyle V(h/k):=\\sum_{m=1}^{k-1}\\{mh/k\\}\\cot(\\pi m/k)=-c_0(\\bar h/k)$, where $\\{y\\}=y-\\lfloor y\\rfloor$ and $h\\bar h\\equiv1\\pmod{k}$ with $1\\leq\\bar h<k$, for coprime $h,k$ CITE{BC}, p. 225. In particular, $V(1/k)=-\\alpha_k$.",
"formula-small": "$P_3(x)=27x^2-1$, $P_4(x)=2x-1$ and $P_6(x)=27x^2-49$, since $\\alpha_3=1/(3\\sqrt3)$, $\\alpha_4=1/2$ and $\\alpha_6=7/(3\\sqrt3)$, with positive square roots."
},
"Similar tables": [
{
"table": "HREF{Bettin_Conrey_cotangent_sums}[Bettin–Conrey cotangent sums]",
"relation": "contains real values $c_0(h/k)$ for reduced numerators; $\\alpha_k=c_0(1/k)$ is a root of $P_k$"
},
{
"table": "HREF{Cyclotomic_polynomials}[Cyclotomic polynomials]",
"relation": "$\\Phi_{\\operatorname{lcm}(4,k)}$ supplies the defining relation for the exact root-of-unity computation"
},
{
"table": "HREF{Gaussian_period_polynomials}[Gaussian period polynomials]",
"relation": "comes from unweighted sums over multiplicative cosets of roots of unity; the cotangent sums here use weighted rational expressions"
},
{
"table": "HREF{Multiplicative_fractional-part_autocorrelation}[Multiplicative fractional-part autocorrelation]",
"relation": "its rational-argument formulas involve the Vasyunin sums related by modular inversion to the roots considered here"
}
],
"Tags": [
"number theory",
"polynomial",
"algebraic"
],
"Keywords": [
"Bettin-Conrey",
"cotangent sum",
"minimal polynomial",
"Estermann",
"Vasyunin"
],
"References": {
"BC": {
"bib": "S. Bettin and B. Conrey, Period functions and cotangent sums, Algebra & Number Theory 7 (2013), no. 1, 215–242, pp. 217, 223–225 and equation (12).",
"doi": "10.2140/ant.2013.7.215",
"arxiv": "1111.0931"
}
},
"Links": {
"DLMFZero": {
"title": "DLMF 25.11.13: Hurwitz zeta at zero",
"url": "https://dlmf.nist.gov/25.11.E13"
}
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "def P(k):\n k = ZZ(k)\n L = lcm(4, k)\n K = CyclotomicField(L)\n z = K.gen()\n w, ii = z^(L//k), z^(L//4)\n alpha = -sum((QQ(m)/k)*ii*(w^m+1)/(w^m-1) for m in range(1,k))\n q = alpha.minpoly()\n p = ZZ['x'](q*q.denominator())\n return ZZ['x'](p / p.content())\nP(5) # 3125*x^4 - 2500*x^2 + 16\n# Default table selection: [P(k) for k in range(3,25)]."
}
},
"Data properties": {
"type": "Z[]",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every denominator $3\\leq k\\leq24$ in increasing order, with $k=2$ explained in Comments. This consecutive prefix covers primes, odd prime powers, powers of two and mixed composites; $24$ is a noncanonical initial cutoff for readable polynomial coefficients, independent of the scalar table’s range",
"rigour details": "Computed by exact arithmetic in $\\mathbb Q(\\zeta_{\\operatorname{lcm}(4,k)})$ from CITE{formula-cyclotomic}, then clearing denominators and taking the primitive positive-leading-coefficient polynomial. Every result has native exact integer coefficients, is irreducible over $\\mathbb Q$, and annihilates the defining algebraic number.\n\nAll 22 polynomials are independently recovered from the finite Estermann double sum CITE{formula-estermann} in $\\mathbb Q[t]/(\\Phi_L)$. Multiplying over all Galois embeddings and removing repeated factors reproduces every primitive polynomial. This uses neither the generator’s cotangent rational expression nor its minimal-polynomial routine; Sage polynomial arithmetic and factorization remain shared dependencies.\n\nA separate direct cotangent evaluation in 512-bit Arb arithmetic checks the 22 values $\\alpha_k=c_0(1/k)$ for $3\\leq k\\leq24$. The balls are finite, have radii below $7.50\\times10^{-152}$ and retain at least 505 bits of relative accuracy, exceeding 100 significant decimal digits. Every polynomial encloses zero at its ball; signs at enclosing endpoints and a nonvanishing derivative isolate the root locally. Exact Galois sign identities and the modular-inverse Vasyunin conversion were checked for all 178 reduced numerators in this prefix. Known polynomials at $k=2,3,4,6$, wrong signs and scales at $k=4$, perturbed polynomial constants, and invalid enclosure controls test the checker. Code, argument-selection rationale and detailed evidence are attached."
},
"Display properties": {
"number-header": "$P_k(x)$"
},
"Numbers": {
"3": {
"number": "27*x^2 - 1"
},
"4": {
"number": "2*x - 1"
},
"5": {
"number": "3125*x^4 - 2500*x^2 + 16"
},
"6": {
"number": "27*x^2 - 49"
},
"7": {
"number": "823543*x^6 - 3176523*x^4 + 1164485*x^2 - 32761"
},
"8": {
"number": "x^2 - 2*x - 1"
},
"9": {
"number": "19683*x^6 - 218700*x^4 + 374544*x^2 - 87616"
},
"10": {
"number": "3125*x^4 - 42500*x^2 + 13456"
},
"11": {
"number": "285311670611*x^10 - 7599665408093*x^8 + 48232034378286*x^6 - 106414058828682*x^4 + 93915840934223*x^2 - 28659972957001"
},
"12": {
"number": "108*x^2 - 540*x - 1"
},
"13": {
"number": "302875106592253*x^12 - 14444812775938220*x^10 + 165293434594885792*x^8 - 579153410913987520*x^6 + 695692861421429248*x^4 - 177281660133706752*x^2 + 12580281941364736"
},
"14": {
"number": "823543*x^6 - 40824203*x^4 + 276530373*x^2 - 404693689"
},
"15": {
"number": "5189853515625*x^8 - 406730742187500*x^6 + 7908285671943750*x^4 - 45384694066687500*x^2 + 79626855512571481"
},
"16": {
"number": "16*x^4 - 96*x^3 - 296*x^2 + 296*x - 47"
},
"17": {
"number": "827240261886336764177*x^16 - 112893965151547134875920*x^14 + 4727753024813233540684960*x^12 - 82906338493360365685937152*x^10 + 688820367557600736961522944*x^8 - 2840249531564954452734660608*x^6 + 5577462430871443287173300224*x^4 - 4234358114360178940687941632*x^2 + 382919844623106684133310464"
},
"18": {
"number": "19683*x^6 - 1898316*x^4 + 5724432*x^2 - 179776"
},
"19": {
"number": "1978419655660313589123979*x^18 - 406617302913343398185744105*x^16 + 27157699406399763382172790060*x^14 - 805542025345850376880968801460*x^12 + 11977774068343796505987136232394*x^10 - 90259024781795510013611794800478*x^8 + 308443463850461353292021497574652*x^6 - 313016379686162738702240753158852*x^4 + 31756876359976677116373457025923*x^2 - 193538908274315778499902389569"
},
"20": {
"number": "50000*x^4 - 100000*x^3 - 4045000*x^2 - 13537000*x - 11569339"
},
"21": {
"number": "262757514534242203161*x^12 - 63946003378396848553404*x^10 + 4259970080104169448019296*x^8 - 91907009021019301722760896*x^6 + 754671721321671090278229504*x^4 - 2034242659804420006291897344*x^2 + 1222627712506387757528387584"
},
"22": {
"number": "285311670611*x^10 - 61964507371789*x^8 + 2293135425894126*x^6 - 17331479384828202*x^4 + 11651869370649871*x^2 - 1503524432944249"
},
"23": {
"number": "20880467999847912034355032910567*x^22 - 8776151484979555027656961006367443*x^20 + 1319096354523530201569325802842763169*x^18 - 96564602008518002905797766511937220445*x^16 + 3896575240489515084067357143510419198406*x^14 - 91351511153257473288004409660027084560030*x^12 + 1249612183179005546708501265830900364640482*x^10 - 9484370490801744714527957665531633229780666*x^8 + 34265552998300853090143591528221619246068451*x^6 - 36083575786735815088879163570137482514515487*x^4 + 8830531480223805954319562728625138088950765*x^2 - 87478235052736252886191075041118786391641"
},
"24": {
"number": "729*x^4 - 14580*x^3 + 51948*x^2 + 61776*x - 264392"
}
}
}