Ramanujan theta function
E355433
The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Ramanujan theta function canonical | 1 |
| Ramanujan’s fifth order mock theta function χ1(q) | 1 |
| Ramanujan’s seventh order mock theta functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410516 — 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 theta function Context triple: [Srinivasa Ramanujan, notableWork, Ramanujan theta function]
-
A.
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.
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B.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
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C.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
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D.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
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E.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
- 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 theta function Target entity description: The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.
-
A.
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.
-
B.
Jacobi triple product
The Jacobi triple product is a fundamental identity in number theory and complex analysis that expresses an infinite product as an infinite sum, playing a key role in the theory of theta functions and q-series.
-
C.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
D.
Ramanujan partition congruences
Ramanujan partition congruences are remarkable number-theoretic results discovered by Srinivasa Ramanujan that describe surprising modular patterns in the partition function, such as specific arithmetic progressions where the number of integer partitions of an integer is divisible by a given prime.
-
E.
Jacobi’s four-square theorem
Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical function
ⓘ
q-series ⓘ special function ⓘ |
| appearsIn |
Ramanujan’s lost notebook
ⓘ
Ramanujan’s lost notebook ⓘ
surface form:
Ramanujan’s notebooks
|
| convergesFor | |q| < 1 ⓘ |
| dependsOn |
complex parameter q
ⓘ
complex variable z ⓘ |
| domain | complex numbers ⓘ |
| field |
complex analysis
ⓘ
modular forms ⓘ number theory ⓘ partition theory ⓘ q-series theory ⓘ |
| generalizationOf | certain Jacobi theta functions ⓘ |
| hasProperty |
admits product expansions
ⓘ
given by an infinite series in powers of q ⓘ holomorphic in z for fixed q with |q| < 1 ⓘ satisfies functional equations ⓘ |
| hasVariable |
q
ⓘ
z ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| introducedIn | early 20th century ⓘ |
| namedAfter | Srinivasa Ramanujan ⓘ |
| relatedTo |
Dedekind eta function
ⓘ
Jacobi theta functions ⓘ
surface form:
Jacobi theta function
mock theta function ⓘ modular equations ⓘ modular forms ⓘ SL(2,ℤ) ⓘ
surface form:
modular group SL(2,Z)
q-Pochhammer symbol ⓘ q-hypergeometric series ⓘ |
| studiedBy |
Bruce C. Berndt
ⓘ
G. N. Watson ⓘ George E. Andrews ⓘ |
| usedIn |
Ramanujan partition congruences
ⓘ
surface form:
Ramanujan’s congruences for partition function
modular transformation formulas ⓘ q-series identities ⓘ theory of mock theta functions ⓘ theory of modular forms ⓘ theory of partitions ⓘ |
| usedToDefine | many mock theta functions ⓘ |
| usedToDerive |
q-series transformation formulas
ⓘ
theta-type identities ⓘ |
| usedToExpress | partition generating functions ⓘ |
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 theta function Description of subject: The Ramanujan theta function is a special type of q-series introduced by Srinivasa Ramanujan that plays a central role in the theory of modular forms, partitions, and mock theta functions.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Ramanujan’s fifth order mock theta function χ1(q)
this entity surface form:
Ramanujan’s seventh order mock theta functions