mock theta functions

E355437

mathematical concept object in number theory q-series

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.

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All labels observed (2)

Label	Occurrences
mock theta functions canonical	2
mock theta conjectures	1

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 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Srinivasa Ramanujan → notableWork → mock theta functions ⓘ

Ramanujan’s lost notebook → contains → mock theta functions ⓘ

Ramanujan’s lost notebook → relatedTo → mock theta functions ⓘ

this entity surface form: mock theta conjectures