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{
"Title": "Jacobi theta₄ constants at singular nomes",
"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 $\\theta_4(0\\mid\\tau_r)=\\theta_4(0,q_r)=1+2\\sum_{n=1}^{\\infty}(-1)^nq_r^{n^2}$ in the DLMF convention CITE{DLMF-definition}.",
"Parameters": {
"r": {
"display": "$r$",
"type": "Z",
"constraints": "$r>0$",
"title": "CM index"
}
},
"Links": {
"DLMF-definition": {
"title": "DLMF §20.2, Definitions and periodic properties",
"url": "https://dlmf.nist.gov/20.2#E4"
},
"DLMF-product": {
"title": "DLMF Equation 20.4.5, Product for the fourth theta constant",
"url": "https://dlmf.nist.gov/20.4#E5"
},
"DLMF-theta3-product": {
"title": "DLMF Equation 20.4.4, Product for the third theta constant",
"url": "https://dlmf.nist.gov/20.4#E4"
},
"DLMF-quartic": {
"title": "DLMF Equation 20.7.5, Jacobi quartic identity",
"url": "https://dlmf.nist.gov/20.7#E5"
},
"DLMF-elliptic": {
"title": "DLMF §20.9, Relations to elliptic integrals",
"url": "https://dlmf.nist.gov/20.9"
},
"DLMF-modular": {
"title": "DLMF §23.15, Modular functions and theta constants",
"url": "https://dlmf.nist.gov/23.15"
},
"DLMF-eta": {
"title": "DLMF §23.17, Special values and infinite products",
"url": "https://dlmf.nist.gov/23.17"
},
"DLMF-Jacobi": {
"title": "DLMF Equation 20.4.6, Jacobi identity",
"url": "https://dlmf.nist.gov/20.4#E6"
}
},
"Tags": [
"special values",
"special functions"
],
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "proven",
"complete": "no",
"complete-note": "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",
"rigour details": "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 426 bits of accuracy, more than 128 decimal digits, and its entire ball lies inside the interval represented by the stored 100-digit decimal. The attached checks compare every row with elementary theta series and the theta-four product at 512 bits, with explicit bounds for the omitted terms and factors. They check the quartic and eta identities, the elliptic relations against T295 and T296, the Ramanujan and Weber relation against T297, and the companion theta-three values against T504. The square-point Gamma identity is an independent special-value check. 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."
},
"Display properties": {
"number-header": "$\\theta_4(0,q_r)$"
},
"Numbers": [],
"Keywords": [
"Jacobi theta4",
"theta four",
"theta_4",
"theta constants",
"singular nomes",
"CM values",
"rectangular CM points",
"complex multiplication"
],
"Comments": {
"comment-lattice": "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. The word singular describes the nome at a quadratic imaginary lattice parameter; it does not assert that the theta constant is algebraic.",
"comment-convention": "Write $m_r=k_r^2=\\lambda(\\tau_r)$ for the elliptic parameter in HREF{Singular_values_of_the_elliptic_modulus}[the singular-modulus table]. The complete elliptic integral HREF{Complete_elliptic_integral_of_the_first_kind_K}[$K(m)$] takes the parameter $m$. DLMF writes this same integral as $K(k)$ with modulus $k=\\sqrt m$ CITE{DLMF-elliptic}. All roots in the formulas are positive real roots.",
"comment-arguments": "HREF{Values_of_the_Jacobi_theta_constant}[The theta-four table at rational nomes] stores the same function with the nome itself as its parameter. Here $r$ indexes the rectangular CM points $\\tau_r=i\\sqrt r$."
},
"Formulas": {
"formula-product": "$\\theta_4(0,q_r)=\\prod_{n=1}^{\\infty}(1-q_r^{2n})(1-q_r^{2n-1})^2>0$ CITE{DLMF-product}.",
"formula-quartic": "$\\theta_3(0,q_r)^4=\\theta_2(0,q_r)^4+\\theta_4(0,q_r)^4$, where $\\theta_2(0,q)=2\\sum_{n=0}^{\\infty}q^{(n+1/2)^2}$ and HREF{Jacobi_theta₃_constants_at_singular_nomes}[$\\theta_3(0,q_r)$] is the companion constant $\\theta_3(0,q)=1+2\\sum_{n=1}^{\\infty}q^{n^2}$ CITE{DLMF-definition} CITE{DLMF-quartic}.",
"formula-elliptic": "For $m_r=(\\theta_2(0,q_r)/\\theta_3(0,q_r))^4$ in HREF{Singular_values_of_the_elliptic_modulus}[the singular-modulus table], $\\theta_4(0,q_r)=(1-m_r)^{1/4}\\sqrt{2K(m_r)/\\pi}$ and $K(1-m_r)=\\sqrt r\\,K(m_r)$. Here $K(m_r)$ is in HREF{Complete_elliptic_integral_of_the_first_kind_at_the_singular_values}[the singular elliptic-integral table], and $K(m)=\\int_0^{\\pi/2}(1-m\\sin^2 t)^{-1/2}\\,dt$ uses the parameter $m$ CITE{DLMF-elliptic} CITE{DLMF-modular}.",
"formula-eta-product": "Dedekind's eta function satisfies $\\eta(\\tau_r)=q_r^{1/12}\\prod_{n=1}^{\\infty}(1-q_r^{2n})=Q_r^{1/24}\\prod_{n=1}^{\\infty}(1-Q_r^n)>0$, where $Q_r=q_r^2=e^{2\\pi i\\tau_r}$ CITE{DLMF-eta}.",
"formula-eta": "$\\theta_2(0,q_r)\\theta_3(0,q_r)\\theta_4(0,q_r)=2\\eta(\\tau_r)^3$ CITE{DLMF-Jacobi} CITE{DLMF-modular}.",
"formula-square": "$\\theta_4(0,e^{-\\pi})=\\theta_2(0,e^{-\\pi})=\\Gamma(1/4)/(2\\pi)^{3/4}$, where HREF{Values_of_the_Gamma_function_at_rational_numbers#1/4}[$\\Gamma(1/4)$] is the Gamma value at one quarter. This follows from $\\lambda(i)=1/2$, $\\eta(i)=\\Gamma(1/4)/(2\\pi^{3/4})$ CITE{DLMF-eta}, and CITE{formula-eta}.",
"formula-weber": "$\\theta_4(0,q_r)=(1-m_r)^{1/4}\\mathfrak f(\\tau_r)^2\\eta(\\tau_r)=\\sqrt2\\,(1-m_r)^{1/4}G_r^2\\eta(\\tau_r)$, where $G_r$ and $\\mathfrak f(\\tau_r)=2^{1/4}G_r$ are HREF{Ramanujan's_class_invariants_G_n_with_Weber's_mathfrak_f_sqrt-n}[Ramanujan and Weber class invariants]. Their defining products give $\\mathfrak f(\\tau_r)^2\\eta(\\tau_r)=\\theta_3(0,q_r)$ CITE{DLMF-theta3-product}; CITE{formula-quartic} gives the remaining factor."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.complex_arb import ComplexBallField\n\nr = 1\nC = ComplexBallField(numberdb.bits(100, losing=96))\ntau = C.gen(0) * C(r).sqrt()\nC(0).jacobi_theta(tau)[3].real()"
}
},
"Similar tables": [
{
"table": "HREF{Jacobi_theta₃_constants_at_singular_nomes}[Jacobi theta-three constants at singular nomes]",
"relation": "The companion constants at the same CM indices satisfy CITE{formula-quartic} and CITE{formula-eta}."
},
{
"table": "HREF{Values_of_the_Jacobi_theta_constant}[$\\theta_4(0,q)$ at rational nomes]",
"relation": "This is the same theta constant indexed by rational nomes; the present table is indexed by the positive integers specifying rectangular CM points."
},
{
"table": "HREF{Singular_values_of_the_elliptic_modulus}[Singular values $k_r$ of the elliptic modulus]",
"relation": "The elliptic parameters $m_r=k_r^2$ determine the ratio $\\theta_4(0,q_r)/\\theta_3(0,q_r)=(1-m_r)^{1/4}$."
},
{
"table": "HREF{Complete_elliptic_integral_of_the_first_kind_at_the_singular_values}[$K(m_r)$ at singular values]",
"relation": "The complete elliptic integrals at the same CM indices give $\\theta_4(0,q_r)$ by CITE{formula-elliptic}."
},
{
"table": "HREF{Complete_elliptic_integral_of_the_first_kind_K}[Complete elliptic integral $K(m)$]",
"relation": "This is the same elliptic integral at rational parameters; its $m=1/2$ entry corresponds to $r=1$ in CITE{formula-elliptic}."
},
{
"table": "HREF{Ramanujan's_class_invariants_G_n_with_Weber's_mathfrak_f_sqrt-n}[Ramanujan and Weber class invariants]",
"relation": "The class invariants at the same lattice parameters determine $\\theta_4(0,q_r)/\\eta(\\tau_r)$ together with $m_r$, by CITE{formula-weber}."
},
{
"table": "HREF{Values_of_the_elliptic_alpha_function}[Elliptic alpha function $\\alpha(r)$]",
"relation": "The elliptic alpha function uses the same CM index $r$, elliptic parameter $m_r$ and integral $K(m_r)$."
}
]
}