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 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410515 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Ramanujan tau function Context triple: [Srinivasa Ramanujan, notableWork, Ramanujan tau function]
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
C.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
D.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Ramanujan tau function Target entity description: 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.
-
A.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
B.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
-
C.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
D.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
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 conjectures ⓘ
linked to:
Ramanujan–Petersson conjecture
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 (proved by Deligne) ⓘ
linked to:
Ramanujan–Petersson conjecture
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Ramanujan tau function Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.