mock theta functions
E355437
Mock theta functions are a class of q-series introduced by Srinivasa Ramanujan that exhibit mysterious modular-like behavior and play a key role in modern number theory and the theory of modular forms.
All labels observed (2)
| Label | Occurrences |
|---|---|
| mock theta functions canonical | 2 |
| mock theta conjectures | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410521 — 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: mock theta functions Context triple: [Srinivasa Ramanujan, notableWork, 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.
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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.
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.
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D.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
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E.
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.
- 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: mock theta functions Target entity description: Mock theta functions are a class of q-series introduced by Srinivasa Ramanujan that exhibit mysterious modular-like behavior and play a key role in modern number theory and the theory of modular forms.
-
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.
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.
-
D.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
object in number theory ⓘ q-series ⓘ |
| appearsIn |
modern research on quantum modular forms
ⓘ
topological invariants of 3-manifolds ⓘ |
| clarifiedBy | Sander Zwegers ⓘ |
| clarifiedIn | early 2000s ⓘ |
| completionRelatedTo | shadow (a unary theta function) ⓘ |
| dependsOn | complex variable q with |q|<1 ⓘ |
| exhibit | modular-like behavior ⓘ |
| field |
modular forms
ⓘ
number theory ⓘ |
| generalizationOf | classical theta functions in a mock sense ⓘ |
| hasExample |
Ramanujan’s fifth order mock theta function χ0(q)
ⓘ
Ramanujan theta function ⓘ
surface form:
Ramanujan’s fifth order mock theta function χ1(q)
Ramanujan theta function ⓘ
surface form:
Ramanujan’s seventh order mock theta functions
Ramanujan’s third order mock theta function f(q) ⓘ |
| hasOrder |
fifth order
ⓘ
seventh order ⓘ third order ⓘ |
| hasProperty |
Fourier coefficients often encode deep arithmetic information
ⓘ
admit non-holomorphic modular completions ⓘ coefficients often have arithmetic significance ⓘ do not transform like classical modular forms ⓘ given by q-series with rapidly convergent coefficients ⓘ have asymptotic behavior similar to modular forms ⓘ now understood within the framework of harmonic Maass forms ⓘ were mysterious for many decades after Ramanujan’s death ⓘ |
| hasVariable | q ⓘ |
| interpretedAs | holomorphic parts of harmonic Maass forms ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| introducedIn |
1920
ⓘ
Ramanujan’s last letter to G. H. Hardy ⓘ |
| relatedTo |
Ramanujan’s lost notebook
ⓘ
harmonic Maass forms ⓘ mock modular forms ⓘ modular forms ⓘ modular transformations ⓘ partition theory ⓘ q-hypergeometric series ⓘ theta functions ⓘ |
| studiedIn | theory of modular forms of half-integral weight ⓘ |
| usedIn |
black hole entropy calculations
ⓘ
combinatorics ⓘ moonshine phenomena ⓘ representation theory ⓘ string theory ⓘ study of partition congruences ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: mock theta functions Description of subject: Mock theta functions are a class of q-series introduced by Srinivasa Ramanujan that exhibit mysterious modular-like behavior and play a key role in modern number theory and the theory of modular forms.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
mock theta conjectures