Jacobi theta₃ constants at singular nomes

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Numbers

$r$

$\theta_3(0,q_r)$

1.086434811213308014575316121510223457070205707245218885920790315981856732267109795960561618489679764

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

1.023523999341005863497798656724971852564914607948784741872151985874134797767846031113022857746876019

1.008666841725232885465487417162683667430282464328049143567062350007344780563743555049119741420707616

1.003734885487739091047679595066953866207994332444519408254958153247325173329563798056949832166444235

1.001779064615381202177110882475934358381789965777873877326019094360370487732371453217220866622341394

1.000909921887256762919286004121566671804588146730301330859241796813958542087950056332754220221829115

1.000491167326285921785729793827326814658794622363230300853476431606564563510667592176560174287530005

1.000276688372042286062204483364677742495813309527477822501068232932101445703821825857121558297222172

1.000161399035140694021502070389399573887508391242375289372799863139144370455187453485128542493007120

10:

1.000096937793894731568940341141091584051440682580458400192358357278780556122688022522470605141859220

11:

1.000059690546777789512636720096492035760291811758082056798821874345795941139126435209508247341058958

12:

1.000037557066632685136150730727319517179060309415959268140310001982398439203354621330178456862954173

Definition

For a positive integer $r$, set $\tau_r=i\sqrt r$ and $q_r=e^{\pi i\tau_r}=e^{-\pi\sqrt r}$, taking the positive square root. The entry is the full Jacobi theta constant $\theta_3(0\mid\tau_r)=\theta_3(0,q_r)=1+2\sum_{n=1}^{\infty}q_r^{n^2}$ in the DLMF convention [1].

Parameters

$r$

— CM index ($r>0$)

Formulas

(1)

$K(m)=\int_0^{\pi/2}(1-m\sin^2 t)^{-1/2}\,dt$ for $0\leq m<1$, with $m$ the elliptic parameter [2].

(2)

$m_r=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$, $K(m_r)=\frac{\pi}{2}\theta_3(0,q_r)^2$, and $K(1-m_r)=\sqrt r\,K(m_r)$, where $K(m_r)$ is stored in the singular elliptic-integral table [2].

(3)

For the theta constants $\theta_2$, $\theta_3$ and $\theta_4$ at a common nome, the Jacobi quartic identity is $\theta_3(0,q_r)^4=\theta_2(0,q_r)^4+\theta_4(0,q_r)^4$ [3].

(4)

$\eta(\tau_r)=q_r^{1/12}\prod_{n=1}^{\infty}(1-q_r^{2n})=Q_r^{1/24}\prod_{n=1}^{\infty}(1-Q_r^n)>0$ [6].

(5)

$\theta_2(0,q_r)\theta_3(0,q_r)\theta_4(0,q_r)=2\eta(\tau_r)^3$ [4] [5].

(6)

$\theta_3(0,e^{-\pi})=\Gamma(1/4)/(\sqrt2\,\pi^{3/4})$, where $\Gamma(1/4)$ is the Gamma value at one quarter. This follows from $\lambda(i)=1/2$, $\eta(i)=\Gamma(1/4)/(2\pi^{3/4})$ [6], and (5).

(7)

$\theta_3(0,q_r)=\mathfrak f(\tau_r)^2\eta(\tau_r)=\sqrt2\,G_r^2\eta(\tau_r)$, where $G_r$ and $\mathfrak f(\tau_r)=2^{1/4}G_r$ are Ramanujan and Weber class invariants. The equality follows by comparing their defining products with the product for $\theta_3$ [7].

Comments

(8)

The lattice $\mathbb Z+\mathbb Z i\sqrt r$ has endomorphism order $\mathbb Z[\sqrt{-r}]$ of discriminant $-4r$. Retaining the square factors of $r$ retains this lattice and order. The word singular describes the nome at a quadratic imaginary lattice parameter; it does not assert that the theta constant is algebraic.

(9)

Write $m_r=k_r^2=\lambda(\tau_r)$ for the elliptic parameter in the singular-modulus table. The complete elliptic integral $K(m)$ takes the parameter $m$. DLMF writes this same integral as $K(k)$ with modulus $k=\sqrt m$ [2].

(10)

The theta constants $\theta_2$ and $\theta_4$ use the same nome convention as $\theta_3$ [1]. For the Dedekind eta product, write $Q_r=q_r^2=e^{2\pi i\tau_r}$ [6].

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()
C(0).jacobi_theta(tau)[2].real()

Links

[1]

DLMF §20.2, Definitions and periodic properties

[2]

DLMF §20.9, Relations to elliptic integrals

[3]

DLMF Equation 20.7.5, Jacobi quartic identity

[4]

DLMF Equation 20.4.6, Jacobi identity

[5]

DLMF Equation 23.15.9, Dedekind eta and theta functions

[6]

DLMF §23.17, Special values and infinite products

[7]

DLMF Equation 20.4.4, Product for the third theta constant

Similar tables

Singular values $k_r$ of the elliptic modulus — The singular moduli satisfy $m_r=k_r^2=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$.

$K(m_r)$ at singular values — The complete elliptic integrals at the same CM indices satisfy $K(m_r)=\pi\theta_3(0,q_r)^2/2$.

Complete elliptic integral $K(m)$ — This is the same elliptic integral at rational parameters; its $m=1/2$ entry corresponds to $r=1$ in (2).

$\theta_4(0,q)$ at rational nomes — That table holds $\theta_4(0,q)$ at rational nomes. (3) relates the theta constants at a common nome; the nomes $q_r$ here are transcendental.

Ramanujan's $G_n$ and Weber's $\mathfrak f(i\sqrt n)$ — The class invariants at the same lattice parameters determine $\theta_3(0,q_r)/\eta(\tau_r)$ by (7).

Elliptic alpha function $\alpha(r)$ — The elliptic alpha function uses the same parameter $m_r$ and the integral $K(m_r)$ in its definition.

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 the full theta constant using Arb complex balls at 429 bits for 100 requested significant digits. Every value retains at least 427 bits of accuracy, more than 128 decimal digits, and its ball lies inside the interval represented by the stored 100-digit decimal.

The attached checks compare every row with elementary theta series at 512 bits, bounding the omitted terms geometrically. They check the quartic and eta identities, using an eta product with an explicit omitted-product bound, and compare the elliptic, Ramanujan and Weber relations with T295, T296 and T297. They also check the elliptic-alpha relation to T299 and the independent Gamma formula at $r=1$. Every comparison requires finite enclosures with radii less than $10^{-90}$ and overlap. Known-value and failure controls precede the comparisons. The checks are repeated on the stored values.