Ramanujan's theta function

DefinitionSummand of Ramanujan's general theta function#✓fully proved

For complex numbers \(a\), \(b\) and an integer \(n\), this is the term \(a^{n(n+1)/2} b^{n(n-1)/2}\) of index \(n\) in Ramanujan's general theta function \(f(a,b) = \sum_{k \in \mathbb{Z}} a^{k(k+1)/2} b^{k(k-1)/2}\).

Both exponents \(n(n+1)/2\) and \(n(n-1)/2\) are nonnegative integers for every integer \(n\), so the term is a product of ordinary powers with natural-number exponents.

noncomputable def ramanujanThetaTerm (a b : ℂ) (n : ℤ) : ℂ :=
  a ^ (n * (n + 1) / 2).toNat * b ^ (n * (n - 1) / 2).toNat

Sources

• the common definition in arXiv:2305.14988, arXiv:2309.06689, arXiv:2312.15513, arXiv:2502.17312, and arXiv:2607.26471.

Used by 5

Used byRamanujan's theta series diverges at \(a = b = 1\) · Ramanujan's general theta function · Raw Ramanujan theta sum · The raw Ramanujan theta sum vanishes at \(a = b = 1\) · Absolute convergence of Ramanujan's general theta series

DefinitionRaw Ramanujan theta sum#✓fully proved

For complex numbers \(a\) and \(b\), the raw Ramanujan theta sum is \(\sum_{n \in \mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}\) when this series is summable, and \(0\) otherwise.

The value \(0\) for a non-summable series is a convention. Ramanujan's general theta function, which requires the series to be summable, is the intended object.

noncomputable def ramanujanThetaRaw (a b : ℂ) : ℂ :=
  ∑' n : ℤ, ramanujanThetaTerm a b n

Uses 1 · Used by 25

Statement usesSummand of Ramanujan's general theta function

Used byRamanujan's theta product identity for \(ab = cd\) · Ramanujan's 5-dissection of Euler's product · Ramanujan's 7-dissection of Euler's product · Cui–Gu \(p\)-dissection of \(f_1\) · Cui–Gu \(p\)-dissection of \(\psi(q)\) · Product forms of Ramanujan's theta functions \(\psi\), \(f\) and \(\chi\) · Ramanujan's Entry 30 (i)–(iii) for the general theta function · Ramanujan's level-10 theta function identities · Ramanujan's general theta function · The raw Ramanujan theta sum vanishes at \(a = b = 1\) · Eta-quotients of Ramanujan's theta functions · Ramanujan's theta function: sum of products when \(ab = cd\) · Ramanujan's theta function: difference of products when \(ab = cd\) · Ramanujan's theta function at \((q, q^5)\) · Ramanujan's theta function at \((q, q^2)\) · Product of the nonic theta functions · Modular relations for the order-eighteen Rogers–Ramanujan–Slater functions · Relations between the order-eighteen and the nonic Rogers–Ramanujan functions · Eisenstein-series formula for \(\varphi(q)\varphi(q^3)\varphi(q^5)\varphi(q^{15})\) · Eisenstein-series formula for \(\varphi(q)^2\varphi(q^{15})^2\) · Eisenstein-series formula for \(\varphi(q)^2\varphi(q^{10})^2\) · A theta identity for \(\varphi(q)^2\varphi(q^{10})^2 + 5\varphi(q^2)^2\varphi(q^5)^2\) · Eisenstein-series formula for \(\varphi(q^3)^2\varphi(q^5)^2\) · Eisenstein-series formula for \(\varphi(q^2)^2\varphi(q^5)^2\) · A theta identity for \(12\varphi(q)^2\varphi(q^{10})^2 + 12\varphi(q^2)^2\varphi(q^5)^2\)

DefinitionRamanujan's general theta function#✓fully proved

For complex numbers \(a\) and \(b\) such that the series \(\sum_{n \in \mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}\) is summable, Ramanujan's general theta function is its sum \(f(a,b) = \sum_{n \in \mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}\).

noncomputable def ramanujanTheta (a b : ℂ)
    (_h : Summable (ramanujanThetaTerm a b)) : ℂ :=
  ramanujanThetaRaw a b

Uses 2

Statement usesRaw Ramanujan theta sum · Summand of Ramanujan's general theta function

TheoremRamanujan's theta series diverges at \(a = b = 1\)#✓fully proved

The series \(\sum_{n \in \mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}\) of Ramanujan's general theta function is not summable at \(a = b = 1\), since every one of its terms equals \(1\).

theorem not_summable_ramanujanThetaTerm_one_one :
    ¬Summable (ramanujanThetaTerm 1 1)

Uses 1 · Used by 1

Statement usesSummand of Ramanujan's general theta function

Used byThe raw Ramanujan theta sum vanishes at \(a = b = 1\)

TheoremThe raw Ramanujan theta sum vanishes at \(a = b = 1\)#✓fully proved

The raw Ramanujan theta sum \(\sum_{n \in \mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}\), which is assigned the value \(0\) whenever the series is not summable, equals \(0\) at \(a = b = 1\), because there every term equals \(1\) and the series is not summable.

theorem ramanujanThetaRaw_one_one : ramanujanThetaRaw 1 1 = 0

Uses 3

Statement usesRaw Ramanujan theta sum

Proof usesRamanujan's theta series diverges at \(a = b = 1\) · Summand of Ramanujan's general theta function

TheoremAbsolute convergence of Ramanujan's general theta series#✓fully proved

For all complex numbers \(a\) and \(b\) with \(|ab| < 1\), the series \(\sum_{n \in \mathbb{Z}} |a^{n(n+1)/2} b^{n(n-1)/2}|\) of the absolute values of the terms of Ramanujan's general theta function converges.

theorem ramanujanTheta_summable (a b : ℂ) (h : ‖a * b‖ < 1) :
    Summable (fun n : ℤ => ‖ramanujanThetaTerm a b n‖)

Sources

• arXiv:2607.26471, Oliver_identities.tex ll. 127-130.

Uses 1

Statement usesSummand of Ramanujan's general theta function

6 fully proved

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