Minimal polynomials of Bettin–Conrey cotangent sums

edit · history · discussion · files · short url · algebraic polynomial number theory

Polynomials

$k$

$P_k(x)$

27*x^2 - 1

2*x - 1

3125*x^4 - 2500*x^2 + 16

27*x^2 - 49

823543*x^6 - 3176523*x^4 + 1164485*x^2 - 32761

x^2 - 2*x - 1

19683*x^6 - 218700*x^4 + 374544*x^2 - 87616

10:

3125*x^4 - 42500*x^2 + 13456

11:

285311670611*x^10 - 7599665408093*x^8 + 48232034378286*x^6 - 106414058828682*x^4 + 93915840934223*x^2 - 28659972957001

12:

108*x^2 - 540*x - 1

13:

302875106592253*x^12 - 14444812775938220*x^10 + 165293434594885792*x^8 - 579153410913987520*x^6 + 695692861421429248*x^4 - 177281660133706752*x^2 + 12580281941364736

14:

823543*x^6 - 40824203*x^4 + 276530373*x^2 - 404693689

15:

5189853515625*x^8 - 406730742187500*x^6 + 7908285671943750*x^4 - 45384694066687500*x^2 + 79626855512571481

16:

16*x^4 - 96*x^3 - 296*x^2 + 296*x - 47

17:

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:

19683*x^6 - 1898316*x^4 + 5724432*x^2 - 179776

19:

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:

50000*x^4 - 100000*x^3 - 4045000*x^2 - 13537000*x - 11569339

21:

262757514534242203161*x^12 - 63946003378396848553404*x^10 + 4259970080104169448019296*x^8 - 91907009021019301722760896*x^6 + 754671721321671090278229504*x^4 - 2034242659804420006291897344*x^2 + 1222627712506387757528387584

22:

285311670611*x^10 - 61964507371789*x^8 + 2293135425894126*x^6 - 17331479384828202*x^4 + 11651869370649871*x^2 - 1503524432944249

23:

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:

729*x^4 - 14580*x^3 + 51948*x^2 + 61776*x - 264392

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 Bettin–Conrey cotangent sum, with $\pi$ positive [1]. 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

$k$

— Denominator ($k\geq2$)

Formulas

(1)

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 $\cot\theta=i(e^{2i\theta}+1)/(e^{2i\theta}-1)$.

(2)

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 $L$.

(3)

$\displaystyle D(0,1/k)=\frac14+\frac{i}{2}\alpha_k=\sum_{r=1}^k\sum_{s=1}^k\left(\frac12-\frac rk\right)\left(\frac12-\frac sk\right)e^{2\pi i rs/k}$. 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 [1], equation (12). The finite expression follows by grouping both divisor variables modulo $k$ and using $\zeta(0,u)=1/2-u$ [2].

(4)

$\displaystyle V(h/k):=\sum_{m=1}^{k-1}\{mh/k\}\cot(\pi m/k)=-c_0(\bar h/k)$, where $\{y\}=y-\lfloor y\rfloor$ and $h\bar h\equiv1\pmod{k}$ with $1\leq\bar h<k$, for coprime $h,k$ [1], p. 225. In particular, $V(1/k)=-\alpha_k$.

(5)

$P_3(x)=27x^2-1$, $P_4(x)=2x-1$ 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.

Comments

(6)

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.

(7)

The omitted case $k=2$ has $\alpha_2=0$ and $P_2(x)=x$.

(8)

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.

(9)

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 (2). Thus numerator $1$ represents the reduced-numerator family up to conjugacy and a sign change.

Programs

(P1)

Sage

def P(k):
    k = ZZ(k)
    L = lcm(4, k)
    K = CyclotomicField(L)
    z = K.gen()
    w, ii = z^(L//k), z^(L//4)
    alpha = -sum((QQ(m)/k)*ii*(w^m+1)/(w^m-1) for m in range(1,k))
    q = alpha.minpoly()
    p = ZZ['x'](q*q.denominator())
    return ZZ['x'](p / p.content())
P(5)  # 3125*x^4 - 2500*x^2 + 16
# Default table selection: [P(k) for k in range(3,25)].

References

[1]

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). (arXiv) (doi)

Links

[2]

DLMF 25.11.13: Hurwitz zeta at zero

Similar tables

Bettin–Conrey cotangent sums — contains real values $c_0(h/k)$ for reduced numerators; $\alpha_k=c_0(1/k)$ is a root of $P_k$

Cyclotomic polynomials — $\Phi_{\operatorname{lcm}(4,k)}$ supplies the defining relation for the exact root-of-unity computation

Gaussian period polynomials — comes from unweighted sums over multiplicative cosets of roots of unity; the cotangent sums here use weighted rational expressions

Multiplicative fractional-part autocorrelation — its rational-argument formulas involve the Vasyunin sums related by modular inversion to the roots considered here

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (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)

How they were obtained:

Computed by exact arithmetic in $\mathbb Q(\zeta_{\operatorname{lcm}(4,k)})$ from (1), 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.

All 22 polynomials are independently recovered from the finite Estermann double sum (3) 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.

A 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.