Statements (24)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:defines |
Fourier coefficients of the discriminant modular form Δ(z)
|
| gptkbp:domain |
positive integers
|
| gptkbp:field |
number theory
|
| gptkbp:firstValue |
τ(1) = 1
|
| gptkbp:form |
Δ(z) = q ∏_{n=1}^∞ (1 - q^n)^{24} = ∑_{n=1}^∞ τ(n)q^n, where q = e^{2πiz}
|
| gptkbp:growthBound |
|τ(p)| ≤ 2p^{11/2} for prime p (Deligne's proof)
|
| https://www.w3.org/2000/01/rdf-schema#label |
Ramanujan tau function
|
| gptkbp:introduced |
gptkb:Srinivasa_Ramanujan
|
| gptkbp:namedAfter |
gptkb:Srinivasa_Ramanujan
|
| gptkbp:property |
τ(p^{k+1}) = τ(p)τ(p^k) - p^{11}τ(p^{k-1}) for prime p, k ≥ 1
τ(mn) = τ(m)τ(n) for coprime m, n |
| gptkbp:relatedTo |
gptkb:Ramanujan's_Delta_function
gptkb:modular_discriminant Hecke operators modular forms |
| gptkbp:satisfies |
gptkb:Ramanujan_conjectures
multiplicative property |
| gptkbp:sequence |
gptkb:A000594_(OEIS)
|
| gptkbp:symbol |
τ(n)
|
| gptkbp:bfsParent |
gptkb:Ramanujan_tau_Dirichlet_series
gptkb:Srinivasa_Ramanujan gptkb:modular_group |
| gptkbp:bfsLayer |
5
|