Values of the Jacobi theta constant $\theta_4(0,q)$

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Numbers

$q$

$\theta_4(0,q)$

1/12:

0.8334297835630038958325661370930199456905125963649336332355687430270559458866502069232443435000068981

1/11:

0.8183184200246960027374176396076481030499040349169078121268321935630692623126979489788826814159479965

1/10:

0.8001999980000001999999998000000000019999999999998000000000000001999999999999999980000000000000000002

1/9:

0.7780826041959810188312347574856528713974090775784935666682489102796533170479134745142500374953928906

1/8:

0.7504882663488459115796484114426614804473060390380495869085701107847114896698560294730153953926265640

1/7:

0.7151186509796368025117958539333924150356969387709416242595227903833795241921855402988117765506442405

1/6:

0.6682096780858585622606834636524879509757692550959403874706155047104141885045160264334271940147744445

2/11:

0.6385488451476958844151005054906132345781759903604787569526510765900006720667908207788279541988748949

1/5:

0.6031989760131071932891137374389533594100093194269488147370746199215417368312953416453486081423271089

2/9:

0.5604302177921309239540981455376128518838922919777067160231460267642301569350519871852687428932087075

1/4:

0.5078048710711282609513233740293515851229745932483153161157165322098564101857772116646898770311327244

3/11:

0.4655935793691523371278127422427349177070600130332747742824285776248078951804048889377125176569300753

2/7:

0.4418738369324509632750150528175875562721796229052388244389537648535537249459508829759475962750697680

3/10:

0.4161606426091747425783006227341330826132572536329205485697821050247987890447966784484593879125090067

1/3:

0.3579231272899599030259112341037224646714049513082803954163690883300487501132413258131403699496238938

4/11:

0.3074753992710443427881226507896631089756825494611182525558282331125383273479507533977852957139168662

3/8:

0.2892577875147163845245444172486432795724538794980206251688251755429727014205900129323330625255598783

2/5:

0.2506765707682886633010584773354776141000802727902141772792608923633519331067701894545775905983581444

5/12:

0.2261928950560674802702310075838481265640795551387850157194514179386226879545463537237741505934042765

3/7:

0.2093560927360337799729536385095343273347350089347093084077452807773258851492096970081366132115008346

4/9:

0.1877993492669674752083623867686985171619456975665243673780121034107725599746426256338707632181921746

5/11:

0.1746357775217822448006250791658700772653314971159208610301930057144421068986958336468514555844368892

1/2:

0.1211242080025805024608492931818675058098582468209605972339011083968975317975043492585129661653781781

6/11:

0.07770240998590889898699010512568912690063571362903806680982269622635798255076873184898904222031685166

5/9:

0.06948977474661941865368232505769363252759011697457684955421173638717609709951734128669997612100840052

4/7:

0.05765179425721636269177679870491231698166597580918999027170873351324825770931432646636815774456167174

7/12:

0.04962636240101650305773390658595701393578019555072604127765111268440240555455603419716848564897815814

3/5:

0.03960316452580476770515475621748900815665704734053002686473260341820335227395832452617004477543280985

5/8:

0.02714046523178527333924621138084208976914352427967847666638016368028317567792920883614573046732927710

7/11:

0.02244995811641888650222743007961771254964557858009638492877427176849411663975363504145818550587360778

2/3:

0.01267038972657299063966092272337368274742829896875516769985713306716097371230379968051922221066701449

7/10:

0.005876410710348859157893764190435159857945745799669721143403329764842247911520809497746706288845012322

5/7:

0.003993779635221903652257572075237356304183846619784937671526105457622235460980204240115723063691463064

8/11:

0.002711057131004092994502138041419146941653936412284706004885753618336631924873192440962719988887836904

3/4:

0.001245309430198100883360550765546521830048696502199748600896527283470485875320670908993661570341287219

7/9:

0.0003851233358290454855786185661850030871683341124082171723415470091441801087115642327793290723280156194

4/5:

0.0001183364390046378921420838455300162320090761397734077763426269088121484615102306895231350330540576303

9/11:

0.00003617237821891970255962414018878658743830577341681677431332066056192326583509454210616209670733291519

5/6:

0.00001100969141047105484491613307015586384386171416672622742164777597391967527628195206042980071903936416

6/7:

0.000001009556680423339826607574576840782044420660344913638397559238397626748496516076764784804938823691512

7/8:

9.159962553050060627076014464728130448771344341406858125626816118607830293120271134852791691534714328e-8

8/9:

8.244420679456433483617401201300823649609182116606161804450523793094741612573331810890560108752319450e-9

9/10:

7.373526938473068047969135475469319468124353000395636092859067650315387147885007239936682990623263839e-10

Definition

For a real nome $0<q<1$, this table stores the Jacobi theta constant $\theta_4(0,q)$, with $q=e^{i\pi\tau}$ and $\theta_4(z,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}\cos(2nz)$ [1] [2].

Parameters

$q$

— real nome ($0<q<1$)

Formulas

(1)

$\theta_4(0,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}$.

(2)

$\theta_3(0,q)^4=\theta_2(0,q)^4+\theta_4(0,q)^4$, where $\theta_2(0,q)=2\sum_{n=0}^{\infty}q^{(n+1/2)^2}$ and $\theta_3(0,q)=\sum_{n\in\mathbb Z}q^{n^2}$ [2] [3].

Comments

(3)

The parameter is the real nome $q$, not the lattice parameter $\tau$. In this rectangular case $0<q<1$, the relation $q=e^{i\pi\tau}$ determines $\tau=\log(q)/(i\pi)$ uniquely [1]. Some modular-form sources write $q$ for $e^{2\pi i\tau}$, which is the square of this nome [5].

(4)

These are theta constants: $z$ is fixed at $0$ and is not a parameter. The value $\theta_1(0,q)$ is $0$ for every $q$, because $\theta_1(z,q)$ is odd in $z$ [2].

Programs

(P1)

Sage

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

CBF = ComplexBallField(numberdb.bits(100, losing=80))
q = QQ(1) / QQ(10)
tau = CBF(0, -1) * CBF(q).log() / CBF.pi()
value = CBF(0).jacobi_theta(tau)[3]
value.real()

Links

[1]

DLMF 20.1: Special notation for theta functions

[2]

DLMF 20.2: Definitions and periodic properties of theta functions

[3]

DLMF 20.7: Identities for theta functions

[4]

DLMF 20.9: Relations of theta functions to other functions

[5]

Wikipedia: Modular form

Similar tables

Values of the elliptic nome $q(m)$ — it stores the nomes $q$ at which these constants are taken, one for each elliptic parameter $m$; at such a nome $\theta_4(0,q)^2/\theta_3(0,q)^2=\sqrt{1-m}$ [4]

Complete elliptic integral of the first kind $K(m)$ — Jacobi's inversion formula gives $K(m)=\frac{\pi}{2}\theta_3(0,q)^2$ when $q$ is the nome of $m$ [4]

Singular values $k_r$ of the elliptic modulus — it stores $k_r=\theta_2(0,q)^2/\theta_3(0,q)^2$ at $q=e^{-\pi\sqrt r}$; at the same nome $\theta_4(0,q)^2/\theta_3(0,q)^2$ is the complementary modulus [4]

Complete elliptic integral of the first kind $K(m_r)$ at the singular values — it stores $K(m_r)=\frac{\pi}{2}\theta_3(0,e^{-\pi\sqrt r})^2$, the singular-nome case of Jacobi's inversion formula [4]

Values of the elliptic alpha function $\alpha(r)$ — its defining formula uses $\theta_4(0,q)$ and its $q$-derivative at the singular nome $q=e^{-\pi\sqrt r}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $\theta_4(0,q)$ at every rational nome $q=a/b$ in lowest terms with $b\leq12$ and $0<q\leq9/10$. The height bound is the one T425 uses, so the two agree wherever both hold a value, but the range is shorter: $\theta_4(0,9/10)$ is already about $7\cdot10^{-10}$ and the next argument up, $q=10/11$, is near $6\cdot10^{-11}$, so beyond $9/10$ the constant is smaller than anything a reader arrives holding)

How they were obtained:

Each value was computed as a Sage complex ball with arb at numberdb.bits(digits, losing=80) bits and compared against the defining $q$-series with an explicit tail bound and against (2) on every row.

The nome was converted to $\tau=\log(q)/(i\pi)$; the imaginary part was checked to contain zero and the real part was returned.