Ramanujan tau function
E355432
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ramanujan tau function canonical | 3 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Fourier coefficient function
ⓘ
arithmetic function ⓘ multiplicative function ⓘ number-theoretic function ⓘ |
| appearsIn | Ramanujan’s paper on highly composite numbers and modular forms ⓘ |
| associatedWith | unique normalized cusp form of weight 12 for SL(2,ℤ) ⓘ |
| codomain | integers ⓘ |
| congruenceProperty |
τ(n) ≡ n^{11} + 1217 n^3 (mod 2^11) for certain n
ⓘ
τ(n) ≡ n^{11} + 5 n^7 (mod 3^6) for certain n ⓘ τ(n) ≡ σ_{11}(n) (mod 691) ⓘ |
| definedAs | Fourier coefficients of the modular discriminant Δ(z) ⓘ |
| domain | positive integers ⓘ |
| eigenformProperty | Hecke eigenvalues equal τ(n) ⓘ |
| generatingFunction | Δ(z) = q ∏_{n≥1} (1 - q^n)^{24} = ∑_{n≥1} τ(n) q^n with q = e^{2πiz} ⓘ |
| growthBound | |τ(p)| ≤ 2 p^{11/2} for prime p ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| LFunction | L(Δ,s) = ∑_{n≥1} τ(n) n^{-s} ⓘ |
| LFunctionType | degree 2 L-function ⓘ |
| multiplicativeProperty | τ(mn) = τ(m)τ(n) for gcd(m,n)=1 ⓘ |
| namedAfter | Srinivasa Ramanujan ⓘ |
| openProblem |
infinitely many n with τ(n) = 0 is unknown
ⓘ
sign changes of τ(n) are not fully understood ⓘ |
| recurrencePrimePowers | τ(p^{k+1}) = τ(p)τ(p^k) - p^{11}τ(p^{k-1}) for prime p and k ≥ 1 ⓘ |
| relatedTo |
Deligne’s proof of the Weil conjectures
ⓘ
Eisenstein series of weight 12 ⓘ Galois representations ⓘ Ramanujan–Petersson conjecture ⓘ
surface form:
Ramanujan conjectures
cusp forms ⓘ discriminant of the elliptic modular function ⓘ modular discriminant Δ(z) ⓘ modular forms ⓘ ℓ-adic Galois representation attached to Δ ⓘ |
| satisfies |
Hecke multiplicativity relations
ⓘ
Ramanujan–Petersson conjecture ⓘ
surface form:
Ramanujan–Petersson conjecture (proved by Deligne)
functional equation of weight 12 cusp form L-function ⓘ |
| studiedIn |
algebraic number theory
ⓘ
analytic number theory ⓘ theory of modular forms ⓘ |
| symbol | τ(n) ⓘ |
| valueAt |
τ(1) = 1
ⓘ
τ(10) = -115920 ⓘ τ(2) = -24 ⓘ τ(3) = 252 ⓘ τ(4) = -1472 ⓘ τ(5) = 4830 ⓘ τ(7) = -16744 ⓘ τ(8) = 84480 ⓘ τ(9) = -113643 ⓘ |
| weight | 12 ⓘ |
| yearIntroduced | 1916 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.