Uniformization radius of the plane punctured at roots of unity

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Numbers

$n$

$\gamma_n$

4.376879230452953277673539881408929086518745445651133444238572421158903876891865895542066299355121726

2.581056539840464461882802458910112905520298584626661427564808095131717393930906636575624451361338591

2.019674148365157845531773667889112082805108492569294377193541486415604379453509967173009744607913438

1.749717157980906734405902335214409329199444833176660875123171601020914712119656307495855518022909074

1.591905460046978768616330485262541624185304322621097376208665097746875062431489105303607486607008791

1.488635396044770999873020427910265408002221529167549955142752240818694384053260838752697078556199069

1.415891655086958047417855396612240436368539309628275051817005849418967809530226552728602287836391166

1.361921424316047568025994077868180793688546465195665696283305291126657990692302901818423658816577607

10:

1.320306382655998889601151309928455097974886068326528729231878215839579975893340479001299824844764798

11:

1.287249164469844054892696818692902623091371253811797121233615702848706153706570744591228618205993716

12:

1.260361328152353988074262112720003615504233702828474356314046303551148759013478001881159219124379144

13:

1.238066235902498402775146076712741955658653152553660438173404788777183689795416352384472985473853972

14:

1.219281573346071137793662764939059275112758527619720841633596651430312245185316210516975653006474571

15:

1.203239911008982994588106074060449231073902770499199122171529440576374319397221522147443539113642511

16:

1.189382068985098885810892757262253787645959913804022963927517025811086790184385169085295393822445508

17:

1.177291031376640961434440383628380450766173187930072599960553846594443230076019157442511599181381167

18:

1.166649505382409024665205808478161094252107966876823859309421118154591873455992091877285679298630016

19:

1.157211817053703129569244569626328494666229889808351274733261472330708723454333711563584229240889169

20:

1.148784798137687760124858419972602681292416692496980214333548003433370264638005157483309125630062505

21:

1.141214479951114727513414973400905189081786503838153738198409523530233337941249673534155397922145518

22:

1.134376636214917801868205924315017403994845170556972098850405636515626392122825995615324390978229199

23:

1.128169936317671360570783938053394597759233697297698233008034434765988072077962463699158889763477581

24:

1.122510905734258945225756513490889933127230944940592044872033180656281447213612515851391187509477778

25:

1.117330160807620390911296117200555410673960129396160067320101420734911556283277179925663678379129647

26:

1.112569557296163017874509666633651585544129328375149664610417279404174251929423599423147231497405674

27:

1.108180004128103995299296937259853329478924685371409500620843451380728624139818203002258072075032147

28:

1.104119768157496977527749900700330160143248883956153871401100687114429628226065015085721550279426169

29:

1.100353145957633228495016210570651702062225076198316597657573504133542674006148790250301918182982166

30:

1.096849413199744608823801508409158538058418665261151720911413066902127329846563313076896426342822418

Definition

Let $\mu_n=\{e^{2\pi i j/n}:0\leq j<n\}$ and $\mathbb D=\{z:|z|<1\}$. The uniformization radius is $\gamma_n=|F_n'(0)|$, where $F_n:\mathbb D\to\mathbb C\setminus\mu_n$ is a universal holomorphic covering map with $F_n(0)=0$ [2].

Parameters

$n$

— number of punctures in the plane ($n\geq2$)

Formulas

(1)

$\displaystyle \gamma_n=16^{1/n}\frac{\Gamma(1+1/(2n))^2\Gamma(1-1/n)}{\Gamma(1-1/(2n))^2\Gamma(1+1/n)}$, where $\Gamma$ is the Gamma function and $16^{1/n}$ is the positive real root. This is equation (5.1.5) of [2], an author version of [1].

(2)

$\displaystyle \gamma_2=\frac{\Gamma(1/4)^4}{4\pi^2}=\frac{4K^2}{\pi}$, where $K=$ $K(1/2)$ uses the elliptic parameter $m=1/2$. The lemniscatic identity follows from [5] and [6].

Comments

(3)

The punctures are the $n$th roots of unity on the unit circle, including $1$. The value $\gamma_n$ is independent of a rotation of the source disk; that rotation may be chosen so that $F_n'(0)>0$.

(4)

The covering map need not be injective. The index $n$ counts punctures in the plane; infinity is an additional puncture in the compactification to the Riemann sphere.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
R = RealBallField(461)
def uniformization_radius(n):
    n = QQ(n)
    if n.denominator() != 1 or n < 2:
        raise ValueError("n must be an integer at least 2")
    a = QQ(1) / n
    h = a / 2
    return (R(16)**R(a) * R(1+h).gamma()**2 * R(1-a).gamma()
            / (R(1-h).gamma()**2 * R(1+a).gamma()))
values = {n: uniformization_radius(n) for n in range(2, 31)}

References

[1]

F. Calegari, V. Dimitrov and Y. Tang, The unbounded denominators conjecture, Journal of the American Mathematical Society 38 (2025), 627–702. (arXiv) (doi)

Links

[2]

Calegari, Dimitrov and Tang: author preprint, Definition 5.1.1 and Theorem 5.1.4

[3]

DLMF 5.7.3: convergent expansion of log Gamma

[4]

DLMF 19.8(i): arithmetic-geometric mean and elliptic integrals

[5]

DLMF 19.20.2: lemniscatic Gamma value

[6]

DLMF 19.25.1: complete elliptic integral as a symmetric integral

Similar tables

Roots of unity — These are the punctures defining each domain.

Values of the Gamma function at rational numbers — The four rational Gamma arguments in the formula are fixed by the puncture count.

Complete elliptic integral of the first kind — The lemniscatic value determines the uniformization radius of the twice-punctured plane.

Data properties

Entries are of type: real number

How well the digits are known: proven

digits: 100 (Unknown key)

Table is complete: no (it holds every $2\leq n\leq30$, with the centre fixed at $0$ and punctures at the $n$th roots of unity on the unit circle. The discrete index is the cyclic symmetry order. This contiguous range includes small prime and composite orders and nonconstructible configurations; $30$ is a noncanonical editorial cutoff giving coverage beyond familiar small symmetries without extending deeply into large orders whose radii approach $1$. There are no independently sampled Gamma arguments or continuous-input grids)

How they were obtained:

Computed with Arb real balls at 461 working bits from exact rational arguments. The 128-bit guard left at least 455 bits of relative accuracy in every entry. Exact rational comparisons of interval endpoints confirm that every serialized 100-significant-digit decimal contains the full computed ball within one unit of its last place.

All 29 values were compared with a separately implemented convergent series in odd zeta values, derived from [3], using 600-bit balls and an explicit bound for the omitted tail after 240 terms. Every serialized interval also contains that independent enclosure. The $n=2$ identity was checked with twelve arithmetic-geometric mean iterations and a rigorous bracket [4]; controls with equal initial arguments were checked first. The numerical evidence, reproducible checks and argument-selection note are attached.