Delta function (Ramanujan's tau function)
GPTKB entity
Statements (30)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:cusp_form
modular group |
| gptkbp:citation |
Modular Forms and Dirichlet Series in Number Theory (Apostol)
OEIS A000594 Ramanujan's original papers |
| gptkbp:degree |
12
|
| gptkbp:discoveredBy |
gptkb:Srinivasa_Ramanujan
|
| gptkbp:domain |
upper half-plane
|
| gptkbp:field |
complex analysis
modular forms number theory |
| gptkbp:first_few_τ(n) |
1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, ...
|
| gptkbp:Fourier_coefficient |
Ramanujan tau function τ(n)
|
| gptkbp:Fourier_expansion |
q∏_{n=1}^∞ (1 - q^n)^{24} = ∑_{n=1}^∞ τ(n)q^n
|
| https://www.w3.org/2000/01/rdf-schema#label |
Delta function (Ramanujan's tau function)
|
| gptkbp:level |
1
|
| gptkbp:multiplicative |
τ(mn) = τ(m)τ(n) for coprime m, n
|
| gptkbp:notation |
Δ
Δ(z) Δ(q) |
| gptkbp:relatedTo |
gptkb:lion
gptkb:Eisenstein_series gptkb:modular_discriminant modular group |
| gptkbp:satisfies |
Ramanujan conjecture (Deligne's theorem)
Δ(az+b/cz+d) = (cz+d)^{12} Δ(z) for SL(2,ℤ) |
| gptkbp:vanishesAt |
infinity
|
| gptkbp:weight |
12
|
| gptkbp:bfsParent |
gptkb:Cusp_forms
|
| gptkbp:bfsLayer |
7
|