Mock theta function
Encyclopedia
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and
a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan
in his last 1920 letter to G. H. Hardy
and in his lost notebook
. Ramanujan's own definition of mock theta functions is notoriously vague, and it was an open problem for many years to find a better definition. This was finally solved by , who discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.
Ramanujan pointed out that they have an asymptotic expansion
at the cusps, similar to that of modular forms of weight 1/2, possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions.
In a notoriously obscure definition, he called functions with similar properties "mock theta functions". No better definition was found for many years, until Zwegers discovered the connection with weak Maass forms.
Ramanujan associated an order to his mock theta functions, which was not clearly defined. Before the work of Zwegers, the orders of known mock theta functions included
Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.
In the next few decades, Ramanujan's mock theta functions were studied by Watson, Andrews, Selberg, Hickerson, Choi, McIntosh, and others, who proved Ramanujan's statements about them and found several more examples and identities. (Most of the "new" identities and examples were already known to Ramanujan and reappeared in his lost notebook.) found that under the action of elements of the modular group
, the order 3 mock theta functions almost transform like modular form
s of weight 1/2 (multiplied by suitable powers of q), except that there are "error terms" in the functional equations, usually given as explicit integrals. However for many years there was no good definition of a mock theta function. This changed in 2001 when Zwegers discovered the relation with non-holomorphic modular forms, Lerch sums, and indefinite theta series.
Zwegers (2002) showed, using the previous work of Watson and Andrews, that the mock theta functions of orders 3, 5, and 7 can be written as the sum of a weak Maass form of weight 1/2 and a function that is bounded along geodesic
s ending at cusps.
The weak Maass form has eigenvalue 3/16 under the hyperbolic Laplacian (the same value as holomorphic modular forms of weight 1/2); however, it increases exponentially fast near cusps, so it does not satisfy the usual growth condition for Maass wave form
s. Zwegers proved this result in three different ways, by relating the mock theta functions to Hecke's theta functions of indefinite lattices of dimension 2, and to Appell–Lerch sums, and to meromorphic Jacobi forms.
Zwegers's fundamental result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2.
This allows one to extend many results about modular forms to mock theta functions. In particular, like modular forms, mock theta functions all lie in certain explicit finite dimensional spaces, which reduces the long and hard proofs of many identities between them to routine linear algebra. For the first time it became possible to produce infinite numbers of examples of mock theta functions; before this work there were only about 50 examples known (most of which were first found by Ramanujan). As further applications of Zwegers's ideas, Kathrin Bringmann
and Ken Ono
showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms , related mock theta functions to Galois representations , and showed that the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by
of and converges to the coefficients . In particular
Mock theta functions have asymptotic expansion
s at cusp
s of the modular group
, acting on the upper half-plane, that resemble those of modular form
s of weight 1/2 with poles at the cusps.
Fix a weight k, usually with 2k integral.
Fix a subgroup Γ of SL2(Z) (or of the metaplectic group
if k is half-integral) and a character ρ of Γ. A modular form f for this character and this group Γ transforms under elements of Γ by
A weak Maass form of weight k is a continuous function on the upper half plane that transforms like a modular form of weight 2 − k and is an eigenfunction of the weight k Laplacian operator, and is called harmonic if its eigenvalue is (1 − k/2)k/2. This is the eigenvalue of holomorphic weight k modular forms, so these are all examples of harmonic weak Maass forms. (A Maass form is a weak Maass form that decreases rapidly at cusps.)
So a harmonic weak Maass form is annihilated by the differential operator
If F is any harmonic weak Maass form then the function g given by
is holomorphic and transforms like a modular form of weight k, though it may not be holomorphic at cusps.
If we can find any other function g* with the same image g, then F − g* will be holomorphic. Such a function is given by inverting the differential operator by integration; for example we can define
where
is essentially the incomplete gamma function
.
The integral converges whenever g has a zero at the cusp i∞, and the incomplete gamma function can be extended by analytic continuation, so this formula can be used to define the holomorphic part g* of F even in the case when g is meromorphic at i∞, though this requires some care if k is 1 or not integral or if n = 0. The inverse of the differential operator is far from unique as we can add any homomorphic function to g* without affecting its image, and as a result the function g* need not be invariant under the group Γ.
The function h = F − g* is called the holomorphic part of F.
A mock modular form is defined to be the holomorphic part h of some harmonic weak Maass form F.
So there is an isomorphism from the space of mock modular forms h to a subspace of the harmonic weak Maass forms.
The mock modular form h is holomorphic but not quite modular, while h + g* is modular but not quite holomorphic. The space of mock modular forms of weight k contains the space of nearly modular forms ("modular forms that may be meromorphic at cusps") of weight k as a subspace. The quotient is (antilinearly) isomorphic to the space of holomorphic modular forms of weight 2 − k. The weight-(2 − k) modular form g corresponding to a mock modular form h is called its shadow. It is quite common for different mock theta functions to have the same shadow. For example, the 10 mock theta functions of order 5 found by Ramanujan fall into two groups of 5, where all the functions in each group have the same shadow (up to multiplication by a constant).
defines a mock theta function as a rational power of q = e2πiτ times a mock modular form of weight 1/2 whose shadow is
a theta series of the form
for a positive rational κ and an odd periodic function ε. (Any such theta series is a modular form of weight 3/2). The rational power of q is a historical accident.
Most mock modular forms and weak Maass forms have rapid growth at cusps. It is common to impose the condition that they grow at most exponentially fast at cusps (which for mock modular forms means they are "meromorphic" at cusps). The space of mock modular forms (of given weight and group) whose growth is bounded by some fixed exponential function at cusps is finite-dimensional.
The Appell–Lerch series is
where
and
The modified series
where
and y = Im(τ) and
satisfies the following transformation properties
In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain non-analytic series added to them.
functions are equal to quotients Θ(τ)/θ(τ) where θ(τ) is a modular form of weight 1/2
and Θ(τ) is a theta function of an indefinite binary quadratic form, and
proved similar results for seventh order mock theta functions. Zwegers showed how to complete the indefinite theta functions to produce real analytic modular forms, and used this to give another proof
of the relation between mock theta functions and weak Maass wave forms.
in terms of quotients of Jacobi's theta functions. Zwegers used this idea to express mock theta functions as Fourier coefficients of meromorphic Jacobi forms.
s of 3-manifolds. related mock theta functions to infinite dimensional Lie superalgebra
s and conformal field theory
. showed that the modular completions of mock modular forms arise as elliptic genera of conformal field theories with continuous spectrum.
Mock theta functions are mock modular forms of weight 1/2 whose shadow is a unary theta function, multiplied by a rational power of q (for historical reasons). Before the work of Zwegers led to a general method for constructing them, most examples were given as basic hypergeometric functions, but this is largely a historical accident, and most mock theta functions have no known simple expression in terms of such functions.
The "trivial" mock theta functions are the (holomorphic) modular forms of weight 1/2, which were classified by , who showed that they could all be written in terms of theta functions of 1-dimensional lattices.
The following examples use the q-Pochhammer symbols
which are defined as:
The function μ was found by Ramanujan in his lost notebook.
These are related to the functions listed in the section on order 8 functions by
. proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appel–Lerch sums. described the asymptotic expansion of their coefficients. related them to
harmonic weak Maass forms. See also
The seven order-3 mock theta functions given by Ramanujan are
, .
.
.
.
.
.
.
The first 4 of these form a group with the same shadow (up to a constant), and so do the last three. More precisely, the functions satisfy the following relations (found by Ramanujan and proved by Watson):
the remaining 5 sixth order mock theta functions in terms of φ and ψ. discovered two more sixth order functions.
The order 6 mock theta functions are:
These three mock theta functions have different shadows, so unlike the case of Ramanujans order 3 and order 5 functions, there are no linear relations between them and ordinary modular forms.
The corresponding weak Maass forms are
where
and
is more or less the complementary error function.
Under the metaplectic group, these 3 functions transform according to a certain 3-dimensional representation of the metaplectic group as follows
, 
, 
In other words, they are the components of a level 1 vector-valued harmonic weak Maass form of weight 1/2.
The two functions V1 and U0 were found earlier by in his lost notebook.
a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...
in his last 1920 letter to G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
and in his lost notebook
Ramanujan's lost notebook
Srinivasa Ramanujan's lost notebook is the manuscript in which Ramanujan, the great Indian mathematician from Cambridge University, recorded the mathematical discoveries of the last year of his life. It was rediscovered by George Andrews in 1976, in a box of effects of G. N. Watson stored at the...
. Ramanujan's own definition of mock theta functions is notoriously vague, and it was an open problem for many years to find a better definition. This was finally solved by , who discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.
History
Ramanujan's 1920 Jan 12 letter to Hardy, reprinted in , listed 17 examples of functions that he called mock theta functions, and his lost notebook contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.)Ramanujan pointed out that they have an asymptotic expansion
Asymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...
at the cusps, similar to that of modular forms of weight 1/2, possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions.
In a notoriously obscure definition, he called functions with similar properties "mock theta functions". No better definition was found for many years, until Zwegers discovered the connection with weak Maass forms.
Ramanujan associated an order to his mock theta functions, which was not clearly defined. Before the work of Zwegers, the orders of known mock theta functions included
- 3, 5, 6, 7, 8, 10.
Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight 1/2 harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.
In the next few decades, Ramanujan's mock theta functions were studied by Watson, Andrews, Selberg, Hickerson, Choi, McIntosh, and others, who proved Ramanujan's statements about them and found several more examples and identities. (Most of the "new" identities and examples were already known to Ramanujan and reappeared in his lost notebook.) found that under the action of elements of the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
, the order 3 mock theta functions almost transform like modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s of weight 1/2 (multiplied by suitable powers of q), except that there are "error terms" in the functional equations, usually given as explicit integrals. However for many years there was no good definition of a mock theta function. This changed in 2001 when Zwegers discovered the relation with non-holomorphic modular forms, Lerch sums, and indefinite theta series.
Zwegers (2002) showed, using the previous work of Watson and Andrews, that the mock theta functions of orders 3, 5, and 7 can be written as the sum of a weak Maass form of weight 1/2 and a function that is bounded along geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s ending at cusps.
The weak Maass form has eigenvalue 3/16 under the hyperbolic Laplacian (the same value as holomorphic modular forms of weight 1/2); however, it increases exponentially fast near cusps, so it does not satisfy the usual growth condition for Maass wave form
Maass wave form
In mathematics, a Maass wave form is a function on the upper half plane that transforms like a modular form but need not be holomorphic. They were first studied by .-Definition:...
s. Zwegers proved this result in three different ways, by relating the mock theta functions to Hecke's theta functions of indefinite lattices of dimension 2, and to Appell–Lerch sums, and to meromorphic Jacobi forms.
Zwegers's fundamental result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2.
This allows one to extend many results about modular forms to mock theta functions. In particular, like modular forms, mock theta functions all lie in certain explicit finite dimensional spaces, which reduces the long and hard proofs of many identities between them to routine linear algebra. For the first time it became possible to produce infinite numbers of examples of mock theta functions; before this work there were only about 50 examples known (most of which were first found by Ramanujan). As further applications of Zwegers's ideas, Kathrin Bringmann
Kathrin Bringmann
Kathrin Bringmann is a number theorist in the University of Cologne, Germany and the University of Minnesota, USA who has made fundamental contributions to the theory of mock theta functions. She has been awarded...
and Ken Ono
Ken Ono
Ken Ono is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, and the fields of interest to Srinivasa Ramanujan...
showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms , related mock theta functions to Galois representations , and showed that the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by
of and converges to the coefficients . In particular
Mock theta functions have asymptotic expansion
Asymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...
s at cusp
Cusp
Cusp may refer to:*Beach cusps, a pointed and regular arc pattern of the shoreline at the beach*Behavioral cusp an important behavior change with far reaching consequences*Cusp catastrophe...
s of the modular group
Modular group
In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
, acting on the upper half-plane, that resemble those of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s of weight 1/2 with poles at the cusps.
Definition
A mock modular form will be defined as the "holomorphic part" of a harmonic weak Maass form.Fix a weight k, usually with 2k integral.
Fix a subgroup Γ of SL2(Z) (or of the metaplectic group
Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers...
if k is half-integral) and a character ρ of Γ. A modular form f for this character and this group Γ transforms under elements of Γ by
A weak Maass form of weight k is a continuous function on the upper half plane that transforms like a modular form of weight 2 − k and is an eigenfunction of the weight k Laplacian operator, and is called harmonic if its eigenvalue is (1 − k/2)k/2. This is the eigenvalue of holomorphic weight k modular forms, so these are all examples of harmonic weak Maass forms. (A Maass form is a weak Maass form that decreases rapidly at cusps.)
So a harmonic weak Maass form is annihilated by the differential operator
If F is any harmonic weak Maass form then the function g given by
is holomorphic and transforms like a modular form of weight k, though it may not be holomorphic at cusps.
If we can find any other function g* with the same image g, then F − g* will be holomorphic. Such a function is given by inverting the differential operator by integration; for example we can define
where
is essentially the incomplete gamma function
Incomplete gamma function
In mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...
.
The integral converges whenever g has a zero at the cusp i∞, and the incomplete gamma function can be extended by analytic continuation, so this formula can be used to define the holomorphic part g* of F even in the case when g is meromorphic at i∞, though this requires some care if k is 1 or not integral or if n = 0. The inverse of the differential operator is far from unique as we can add any homomorphic function to g* without affecting its image, and as a result the function g* need not be invariant under the group Γ.
The function h = F − g* is called the holomorphic part of F.
A mock modular form is defined to be the holomorphic part h of some harmonic weak Maass form F.
So there is an isomorphism from the space of mock modular forms h to a subspace of the harmonic weak Maass forms.
The mock modular form h is holomorphic but not quite modular, while h + g* is modular but not quite holomorphic. The space of mock modular forms of weight k contains the space of nearly modular forms ("modular forms that may be meromorphic at cusps") of weight k as a subspace. The quotient is (antilinearly) isomorphic to the space of holomorphic modular forms of weight 2 − k. The weight-(2 − k) modular form g corresponding to a mock modular form h is called its shadow. It is quite common for different mock theta functions to have the same shadow. For example, the 10 mock theta functions of order 5 found by Ramanujan fall into two groups of 5, where all the functions in each group have the same shadow (up to multiplication by a constant).
defines a mock theta function as a rational power of q = e2πiτ times a mock modular form of weight 1/2 whose shadow is
a theta series of the form
for a positive rational κ and an odd periodic function ε. (Any such theta series is a modular form of weight 3/2). The rational power of q is a historical accident.
Most mock modular forms and weak Maass forms have rapid growth at cusps. It is common to impose the condition that they grow at most exponentially fast at cusps (which for mock modular forms means they are "meromorphic" at cusps). The space of mock modular forms (of given weight and group) whose growth is bounded by some fixed exponential function at cusps is finite-dimensional.
Appell–Lerch sums
Appell–Lerch sums were first studied by and . Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.The Appell–Lerch series is
where
and
The modified series
where
and y = Im(τ) and
satisfies the following transformation properties
In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain non-analytic series added to them.
Indefinite theta series
showed that several of Ramanujan’s fifth order mock thetafunctions are equal to quotients Θ(τ)/θ(τ) where θ(τ) is a modular form of weight 1/2
and Θ(τ) is a theta function of an indefinite binary quadratic form, and
proved similar results for seventh order mock theta functions. Zwegers showed how to complete the indefinite theta functions to produce real analytic modular forms, and used this to give another proof
of the relation between mock theta functions and weak Maass wave forms.
Meromorphic Jacobi forms
observed that some of Ramanujan's fifth order mock theta functions could be expressedin terms of quotients of Jacobi's theta functions. Zwegers used this idea to express mock theta functions as Fourier coefficients of meromorphic Jacobi forms.
Applications
related mock theta functions to quantum invariantQuantum invariant
In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.-List of invariants:*Finite type invariant*Kontsevich invariant*Kashaev's invariant...
s of 3-manifolds. related mock theta functions to infinite dimensional Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
s and conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
. showed that the modular completions of mock modular forms arise as elliptic genera of conformal field theories with continuous spectrum.
Examples
- Any modular form of weight k (possibly only meromorphic at cusps) is a mock modular form of weight k with shadow 0.
- The quasimodular Eisenstein series
-
- of weight 2 and level 1 is a mock modular form of weight 2, with shadow a constant. This means that
- transforms like a modular form of weight 2 (where τ = x + iy).
- The function studied by with Fourier coefficients that are Hurwitz class numbers H(N) of imaginary quadratic fields is a mock modular form of weight 3/2, level 4 and shadow ∑ q n2. The corresponding weak Maass wave form is
-
- where
- and y = Im(τ), q = e2πiτ.
Mock theta functions are mock modular forms of weight 1/2 whose shadow is a unary theta function, multiplied by a rational power of q (for historical reasons). Before the work of Zwegers led to a general method for constructing them, most examples were given as basic hypergeometric functions, but this is largely a historical accident, and most mock theta functions have no known simple expression in terms of such functions.
The "trivial" mock theta functions are the (holomorphic) modular forms of weight 1/2, which were classified by , who showed that they could all be written in terms of theta functions of 1-dimensional lattices.
The following examples use the q-Pochhammer symbols
which are defined as:Order 2
Some order 2 mock theta functions were studied by . The function μ was found by Ramanujan in his lost notebook.
These are related to the functions listed in the section on order 8 functions by
Order 3
Ramanujan mentioned four order-3 mock theta functions in his letter to Hardy, and listed a further three in his lost notebook, which were rediscovered by G. N. WatsonG. N. Watson
Neville Watson was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's A Course of Modern Analysis produced the classic “Whittaker & Watson” text...
. proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appel–Lerch sums. described the asymptotic expansion of their coefficients. related them to
harmonic weak Maass forms. See also
The seven order-3 mock theta functions given by Ramanujan are
, . . . . . . .The first 4 of these form a group with the same shadow (up to a constant), and so do the last three. More precisely, the functions satisfy the following relations (found by Ramanujan and proved by Watson):
Order 5
Ramanujan wrote down ten mock theta functions of order 5 in his 1920 letter to Hardy, and stated some relations between them that were proved by . In his lost notebook he stated some further identities relating these functions, equivalent to the mock theta conjectures , that were proved by . found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight 1/2. Order 6
wrote down seven mock theta functions of order 6 in his lost notebook, and stated 11 identities between them, which were proved in . Two of Ramanujan's identities relate φ and ψ at various arguments, four of them express φ and ψ in terms of Appell–Lerch series, and the last five identities expressthe remaining 5 sixth order mock theta functions in terms of φ and ψ. discovered two more sixth order functions.
The order 6 mock theta functions are:
Order 7
Ramanujan gave three mock theta functions of order 7 in his 1920 letter to Hardy. They were studied by , who found asymptotic expansion for their coefficients, and in . found representations of many of these functions as the quotients of indefinite theta series by modular forms of weight 1/2. described their modular transformation properties.
These three mock theta functions have different shadows, so unlike the case of Ramanujans order 3 and order 5 functions, there are no linear relations between them and ordinary modular forms.
The corresponding weak Maass forms are
where
and
is more or less the complementary error function.
Under the metaplectic group, these 3 functions transform according to a certain 3-dimensional representation of the metaplectic group as follows
, , In other words, they are the components of a level 1 vector-valued harmonic weak Maass form of weight 1/2.
Order 8
found eight mock theta functions of order 8. They found 5 linear relations involving them, and expressed 4 of the functions as Appell–Lerch sums, and described their transformations under the modular group.The two functions V1 and U0 were found earlier by in his lost notebook.
Order 10
listed four order-10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by .
External links
- International Conference: Mock theta functions and applications 2009
- Papers on mock theta functions by George Andrews
- Papers on mock theta functions by Kathrin BringmannKathrin BringmannKathrin Bringmann is a number theorist in the University of Cologne, Germany and the University of Minnesota, USA who has made fundamental contributions to the theory of mock theta functions. She has been awarded...
- Papers on mock theta functions by Ken OnoKen OnoKen Ono is an American mathematician who specializes in number theory, especially in integer partitions, modular forms, and the fields of interest to Srinivasa Ramanujan...
- Papers on mock theta functions by Sander Zwegers