Dedekind eta values at rectangular CM points

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Numbers

$r$

$\eta(i\sqrt r)$

0.7682254223260566590025941795761806445178669144648050146767028241436309867120692077191751030490062522

comment: The square lattice point $\tau_1=i$. Its Gamma expression is (6).

0.6904728558654830349109075372698227329278805227121489706328810274774304558382182445909821424484822558

0.6354202931103006378376900154813439543224708874388329265033017510107416920536796228062515590140912145

0.5923827813324158852903633744919953727615299932057706652342899396271762351255539703205501769417891483

0.5568819397647346306631014064179668836399719083637298941950482152903154185289024911556289412432344275

0.5266204903253668558743616432036604481586047534507005439241949362039939617540138683965907473789921876

0.5002455836631429608928648969136634073293973111816151504267382158032822817553337960882800871437650023

0.4768847131419665144411510967510949989657453029385503473754239916082030929912777502211514979222448308

0.4559381247967392208640737462862162095241775312712659042168601191147188516896137804827007213232757532

10:

0.4369736610795143656574302495179474865778914293750789328293955742251312159437444352344681822772516668

11:

0.4196684263497416113065435198463617881982963545809119979935943367050904178494279001598892359150349701

12:

0.4037741134676644344791373444261389678998379094788831566734536501605898971263445571350179884109382912

Definition

For a positive integer $r$, the entry is Dedekind's eta value $\eta(i\sqrt r)$, with the positive square root and the normalisation $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi in\tau})$ for $\Im\tau>0$ [1].

Parameters

$r$

— CM index ($r>0$)

Formulas

(1)

$\eta(\tau_r)=q_r^{1/12}\prod_{n=1}^{\infty}(1-q_r^{2n})=Q_r^{1/24}\prod_{n=1}^{\infty}(1-Q_r^n)>0$ [1].

(2)

In the DLMF convention, $\theta_2(0,q)=2\sum_{n=0}^{\infty}q^{(n+1/2)^2}$, $\theta_3(0,q)=1+2\sum_{n=1}^{\infty}q^{n^2}$ and $\theta_4(0,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}$ for $0<q<1$ [2].

(3)

$2\eta(\tau_r)^3=\theta_2(0,q_r)\theta_3(0,q_r)\theta_4(0,q_r)$, where $\theta_3(0,q_r)$ and $\theta_4(0,q_r)$ are the companion theta constants [3] [4].

(4)

$\eta(\tau_r)=2^{-1/3}[m_r(1-m_r)]^{1/12}\sqrt{2K(m_r)/\pi}$, where $m_r=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$ is in the singular-modulus table and $K(m_r)$ satisfies $K(m_r)=\pi\theta_3(0,q_r)^2/2$. This follows from (3) and the theta expressions for the modulus, complementary modulus and elliptic integral [4] [5].

(5)

$\eta(\tau_r)=\theta_3(0,q_r)/\mathfrak f(\tau_r)^2=\theta_3(0,q_r)/(\sqrt2\,G_r^2)$, where $G_r=2^{-1/4}q_r^{-1/24}\prod_{n\geq1}(1+q_r^{2n-1})$ and $\mathfrak f(\tau_r)=2^{1/4}G_r$ are Ramanujan and Weber class invariants. The relation follows by comparing their defining products with the product for $\theta_3$ [6].

(6)

$\eta(i)=\Gamma(1/4)/(2\pi^{3/4})$, where $\Gamma(1/4)$ is the Gamma value at one quarter [1].

(7)

$\eta(i/\sqrt r)=r^{1/4}\eta(i\sqrt r)$, a special case of $\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau)$ [7].

Comments

(8)

The lattice $\mathbb Z+\mathbb Z i\sqrt r$ has complex multiplication by the order $\mathbb Z[\sqrt{-r}]$, of discriminant $-4r$. Retaining square factors of $r$ retains this lattice and order. These are rectangular CM points, rather than a list of all CM points or all ideal classes.

(9)

Write $\tau_r=i\sqrt r$, $q_r=e^{\pi i\tau_r}=e^{-\pi\sqrt r}$ and $Q_r=q_r^2=e^{2\pi i\tau_r}$. The theta convention uses $q_r$, whereas the eta product is often written using $Q_r$. At the rectangular CM points $\tau_r$, the eta value is positive real and every root takes its positive real value. For general $\Im\tau>0$, the square root in (7) is principal [7].

(10)

Write $m_r=k_r^2=\lambda(\tau_r)$ for the elliptic parameter in the singular-modulus table. The complete elliptic integral $K(m)=\int_0^{\pi/2}(1-m\sin^2 t)^{-1/2}\,dt$ takes the parameter $m$. DLMF writes this same integral as $K(k)$ with modulus $k=\sqrt m$ [5].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField

r = 1
C = ComplexBallField(numberdb.bits(100, losing=96))
tau = C.gen(0) * C(r).sqrt()
tau.modular_eta().real()

Links

[1]

DLMF §23.17, Special values and infinite products

[2]

DLMF §20.2, Theta functions and their Fourier series

[3]

DLMF Equation 20.4.6, Jacobi identity

[4]

DLMF §23.15, Modular functions and theta constants

[5]

DLMF §20.9, Relations to elliptic integrals

[6]

DLMF Equation 20.4.4, Product for the third theta constant

[7]

DLMF §23.18, Modular transformations of Dedekind eta

Similar tables

Jacobi theta-three constants at singular nomes — The theta-three constants at the same CM indices enter (3) and (5).

Jacobi theta-four constants at singular nomes — The theta-four constants at the same CM indices enter (3).

Singular values of the elliptic modulus — The parameters $m_r=k_r^2$ determine the algebraic factor $[m_r(1-m_r)]^{1/12}$ in (4).

$K(m_r)$ at singular values — The elliptic integrals at the same CM indices give the period factor in (4).

Complete elliptic integral $K(m)$ — The same elliptic integral at rational parameters includes $m=1/2$, which corresponds to $r=1$ in (4).

Ramanujan and Weber class invariants — These class invariants at the same lattice parameters give the ratio $\theta_3(0,q_r)/\eta(\tau_r)$ by (5).

Elliptic alpha function $\alpha(r)$ — The elliptic alpha function uses the same CM index $r$, elliptic parameter $m_r$ and integral $K(m_r)$.

Gamma values at rational arguments — The Gamma value at $1/4$ gives the square lattice eta value by (6).

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds every integer $1\leq r\leq12$, the first small rectangular CM cases, including the square lattice and the square-index changes of scale)

How they were obtained:

The generator evaluates the positive square root in $\tau_r$ and Dedekind eta using Arb complex balls at 429 bits for 100 requested significant digits. Every value retains at least 425 bits of accuracy, and its entire ball lies inside the interval represented by the stored 100-digit decimal. The independent checks use real balls at 512 bits.

The eta product is truncated after $M=256$ factors; the omitted product lies between $1-Q^{M+1}/(1-Q)$ and $1$. The theta series are truncated at $N=12$, with explicit geometric bounds for their tails. The attached checks compare all twelve rows with these independent enclosures, the theta product identity, modular reciprocity, and the elliptic and class-invariant relations using T295, T296, T297, T504 and T505. They also compare the elliptic-alpha relation with T299 and the independent Gamma formula at $r=1$. Every comparison requires finite enclosures with radii less than $10^{-90}$ and overlap. Known-value controls and controls that must fail precede the comparisons. The checks are repeated on the stored values.