View source for Mock modular form

{{Short description|Complex-differentiable part of a Maass wave function}}
{{CS1 config|mode=cs2}}
{{Use shortened footnotes|date=May 2021}}
In [[mathematics]], a '''mock modular form''' is the [[Holomorphic function|holomorphic]] part of a harmonic weak [[Maass wave form|Maass form]], and a '''mock theta function''' is essentially a mock modular form of weight {{sfrac|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 [[Ramanujan's lost notebook|lost notebook]]. [[Sander Zwegers]] discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.{{sfn|Zwegers|2001}}{{sfn|Zwegers|2002}}

==History==
{{quote box
 | align=right
 | width=33%
 | quote="Suppose there is a function in the Eulerian form and suppose that all or an infinity of points are exponential singularities, and also suppose that at these points the asymptotic form closes as neatly as in the cases of (A) and (B). The question is: Is the function taken the sum of two functions one of which is an ordinary ''&theta;''-function and the other a (trivial) function which is O(1) at ''all'' the points ''e''<sup>2''m''{{pi}}''i''/''n''</sup>? ... When it is not so, I call the function a Mock &theta;-function."
 | source=Ramanujan's original definition of a mock theta function{{sfn|Ramanujan|2000|loc=Appendix II}}
}}
Ramanujan's 12 January 1920 letter to Hardy{{sfn|Ramanujan|2000|loc=Appendix II}} listed 17 examples of functions that he called mock theta functions, and his [[lost notebook]]{{sfn|Ramanujan|1988}} 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]] at the cusps, similar to that of modular forms of weight {{sfrac|1|2}}, possibly with poles at cusps, but cannot be expressed in terms of "ordinary" [[theta function]]s. He called functions with similar properties "mock theta functions". Zwegers later discovered the connection of the mock theta function 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 a Dirichlet character|conductor]] of the [[Nebentypus character]] of the weight {{sfrac|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.) In 1936, Watson 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 {{sfrac|1|2}} (multiplied by suitable powers of ''q''), except that there are "error terms" in the functional equations, usually given as explicit integrals.{{sfn|Watson|1936}}  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 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 {{sfrac|1|2}} and a function that is bounded along [[geodesic]]s ending at cusps.{{sfn|Zwegers|2002}} The weak Maass form has [[eigenvalue]] {{sfrac|3|16}} under the [[hyperbolic Laplacian]] (the same value as holomorphic modular forms of weight {{sfrac|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 {{sfrac|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 number 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 {{sfrac|3|2}} harmonic weak Maass forms{{sfn|Bringmann|Folsom|Ono|2009}} and showed that the asymptotic series for coefficients of the order 3 mock theta function ''f''(''q'') studied by [[George Andrews (mathematician)|George Andrews]]{{sfn|Andrews|1966}} and Leila Dragonette{{sfn|Dragonette|1952}} converges to the coefficients.{{sfn|Bringmann|Ono|2006}} In particular Mock theta functions have [[asymptotic expansion]]s at [[cusp form|cusps]] of the [[modular group]], acting on the [[upper half-plane]], that resemble those of [[modular form]]s of weight {{sfrac|1|2}} with poles at the cusps.

==Definition==
A mock modular form will be defined as the "holomorphic part" of a [[harmonic Maass form|harmonic weak Maass form]].

Fix a weight ''k'', usually with 2''k'' integral.
Fix a subgroup Γ of [[special linear group|SL<sub>2</sub>(''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

:<math>f\left(\frac{a\tau + b}{c\tau + d}\right) = \rho{
  \begin{pmatrix}
    a & b  \\
    c & d
  \end{pmatrix}
}(c\tau + d)^kf(\tau)</math>

A '''[[harmonic Maass form|weak Maass form]]''' of weight ''k'' is a [[continuous function]] on the upper half plane that transforms like a modular form of weight ''k'' and is an eigenfunction of the weight ''k'' Laplacian operator, and is called '''harmonic''' if its eigenvalue is {{nowrap|({{sfrac|1 &minus; ''k''|2}}){{sfrac|''k''|2}}}}.{{sfn|Bruinier|Funke|2004}} 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

:<math>\frac{\partial}{\partial \tau}y^k\frac{\partial}{\partial \overline\tau}</math>

If ''F'' is any harmonic weak Maass form then the function ''g'' given by

:<math>g = y^k\frac{\partial \overline{F}}{\partial \tau} = \sum_n b_nq^n</math>

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''<sup>*</sup> with the same image ''g'', then ''F''&nbsp;&minus;&nbsp;''g''<sup>*</sup> will be holomorphic. Such a function is given by inverting the differential operator by integration; for example we can define
:<math>g^*(\tau) = \left(\frac{i}{2}\right)^{k - 1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz= \sum_nn^{k - 1}\overline {b_n}\beta_k(4ny)q^{-n + 1}</math>

where
:<math>\displaystyle \beta_k(t) = \int_t^\infty u^{-k} e^{-\pi u} \,du</math>

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''<sup>*</sup> 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''&nbsp;=&nbsp;0. The inverse of the differential operator is far from unique as we can add any homomorphic function to ''g''<sup>*</sup> without affecting its image, and as a result the function ''g''<sup>*</sup> need not be invariant under the group Γ. The function ''h'' = ''F''&nbsp;&minus;&nbsp;''g''<sup>*</sup> 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''&nbsp;+&nbsp;''g''<sup>*</sup> 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&nbsp;&minus;&nbsp;''k''. The weight-(2&nbsp;&minus;&nbsp;''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).

[[Don Zagier]]{{sfn|Zagier|2007}} defines a '''mock theta function''' as a rational power of ''q''&nbsp;=&nbsp;e<sup>2{{pi}}''i''{{tau}}</sup> times a mock modular form of weight {{sfrac|1|2}} whose shadow is a theta series of the form

:<math>\sum_{n\in Z}\varepsilon(n)nq^{\kappa n^2}</math>

for a positive rational ''κ'' and an odd periodic function ''&epsilon;''. (Any such theta series is a modular form of weight {{sfrac|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, a generalization of [[Lambert series]], were first studied by [[Paul Émile Appell]]{{sfn|Appell|1884}} and [[Mathias Lerch]].{{sfn|Lerch|1892}} 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
:<math>\mu(u, v; \tau) = \frac{a^\frac{1}{2}}{\theta(v; \tau)}\sum_{n\in Z}\frac{(-b)^nq^{\frac{1}{2}n(n + 1)}}{1 - aq^n}</math>

where
:<math>\displaystyle q = e^{2\pi i \tau},\quad a = e^{2\pi i u},\quad b = e^{2\pi i v}</math>

and
:<math>\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n + \frac{1}{2}}q^{\frac{1}{2}\left(n + \frac{1}{2}\right)^2}.</math>

The modified series
:<math>\hat\mu(u, v; \tau) = \mu(u, v; \tau) - \frac{1}{2}R(u - v; \tau)</math>

where
:<math>R(z;\tau) = \sum_{\nu\in Z + \frac{1}{2}}(-1)^{\nu - \frac{1}{2}}\left({\rm sign}(\nu) - E\left[\left(\nu + \frac{\Im(z)}{y}\right)\sqrt{2y}\right]\right)e^{-2\pi i \nu z}q^{-\frac{1}{2}\nu^2}</math>

and ''y'' = Im({{tau}}) and
:<math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>

satisfies the following transformation properties

:<math>\begin{align}
                      \hat\mu(u + 1, v; \tau) &= a^{-1}bq^{-\frac{1}{2}}\hat\mu(u + \tau, v; \tau) \\
                                              & {} = -\hat\mu(u, v; \tau) \\
  e^{\frac{2}{8}\pi i}\hat\mu(u, v; \tau + 1) &= \hat\mu(u,v;\tau) \\
                                              & {} = -\left(\frac{\tau}{i}\right)^{-\frac{1}{2}}e^{\frac{\pi i}{\tau} (u - v)^2}\hat\mu\left(\frac{u}{\tau},\frac{v}{\tau};-\frac{1}{\tau}\right).
\end{align}</math>

In other words, the modified Appell–Lerch series transforms like a modular form with respect to {{tau}}. 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==
George Andrews{{sfn|Andrews|1986}} showed that several of Ramanujan's fifth order mock theta functions are equal to quotients {{sfrac|Θ({{tau}})|''θ''({{tau}})}} where ''θ''({{tau}}) is a modular form of weight {{sfrac|1|2}} and Θ({{tau}}) is a theta function of an '''indefinite'''  binary quadratic form, and Dean Hickerson{{sfn|Hickerson|1988b}} 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==
George Andrews{{sfn|Andrews|1988}} observed that some of Ramanujan's fifth order mock theta functions could be expressed 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.

==Applications==
* Ruth Lawrence and [[Don Zagier]] related mock theta functions to [[quantum invariant]]s of 3-manifolds. {{sfn|Lawrence|Zagier|1999}}
* A. M. Semikhatov, A. Taormina, and I. Yu Tipunin related mock theta functions to infinite-dimensional [[Lie superalgebra]]s and [[two-dimensional conformal field theory]].{{sfn|Semikhatov|Taormina|Tipunin|2005}}
* J. Troost showed that the modular completions of mock modular forms arise as elliptic genera of conformal field theories with continuous spectrum.{{sfn|Troost|2010}}
* Mock theta functions appear in the theory of [[umbral moonshine]].
* [[Atish Dabholkar]], Sameer Murthy, and [[Don Zagier]] showed that mock modular forms are related to the degeneracies of quantum black holes in ''N''=4 string theories.{{sfn|Dabholkar|Murthy|Zagier|2012}}

==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
::<math>\displaystyle E_2(\tau) = 1-24\sum_{n>0}\sigma_1(n)q^n</math>
:of weight 2 and level 1 is a mock modular form of weight 2, with shadow a constant. This means that
::<math>\displaystyle E_2(\tau) -\frac{3}{\pi y}</math>
:transforms like a modular form of weight 2 (where {{tau}} = ''x''&nbsp;+&nbsp;''iy'').

* The function studied by Don Zagier{{sfn|Zagier|1975}}{{sfn|Hirzebruch|Zagier|1976|loc=2.2}} with Fourier coefficients that are Hurwitz class numbers ''H''(''N'') of imaginary quadratic fields is a mock modular form of weight {{sfrac|3|2}}, level 4 and shadow Σ&nbsp;''q''<sup>&nbsp;''n''<sup>2</sup></sup>. The corresponding weak Maass wave form is
::<math>F(\tau) = \sum_NH(N)q^n + y^{-1/2}\sum_{n\in Z}\beta(4\pi n^2y)q^{-n^2}</math>
:where
::<math>\beta(x) = \frac{1}{16\pi}\int_1^\infty u^{-3/2}e^{-xu}du</math>
:and ''y''&nbsp;=&nbsp;Im({{tau}}), ''q''&nbsp;=&nbsp;e<sup>2{{pi}}''i''{{tau}} </sup>.

Mock theta functions are mock modular forms of weight {{sfrac|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 function]]s, 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 {{sfrac|1|2}}, which were classified by Serre and Stark,{{sfn|Serre|Stark|1977}} who showed that they could all be written in terms of theta functions of 1-dimensional lattices.

The following examples use the [[q-Pochhammer symbol]]s (''a'';''q'')<sub>''n''</sub> which are defined as:

:<math>(a;q)_n = \prod_{0\le j<n}(1-aq^j) = (1-a)(1-aq)\cdots(1-aq^{n-1}).</math>

===Order 2===
Some order 2 mock theta functions were studied by McIntosh.{{sfn|McIntosh|2007}}
:<math>A(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2}(-q;q^2)_n}{(q;q^2)^2_{n+1}} = \sum_{n\ge 0} \frac{q^{n+1}(-q^2;q^2)_n}{(q;q^2)_{n+1}}</math> {{OEIS|id=A006304}}
:<math>B(q) = \sum_{n\ge 0} \frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)^2_{n+1}} = \sum_{n\ge 0} \frac{q^{n}(-q;q^2)_n}{(q;q^2)_{n+1}}</math>  {{OEIS|id=A153140}}
:<math>\mu(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q^2;q^2)^2_{n}} </math>  {{OEIS|id=A006306}}

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
:<math> U_0(q) - 2U_1(q) = \mu(q)</math>
:<math> V_0(q) - V_0(-q) = 4qB(q^2)</math>
:<math> V_1(q) + V_1(-q) = 2A(q^2)</math>

===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. Watson]].{{sfn|Watson|1936}} The latter proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appell–Lerch sums. Dragonette{{sfn|Dragonette|1952}} described the asymptotic expansion of their coefficients. Zwegers{{sfn|Zwegers|2001}} related them to harmonic weak Maass forms. See also the monograph by Nathan Fine.{{sfn|Fine|1988}}

The seven order-3 mock theta functions given by Ramanujan are

:<math>
f(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q; q)_n^2}  = \frac{2}{\prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^nq^{n(3n+1)/2}}{1+q^n}
</math>, {{OEIS|id=A000025}}.

:<math>
\phi(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q^2;q^2)_n} = \frac{1}{\prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^n(1+q^n)q^{n(3n+1)/2}}{1+q^{2n}}
</math> {{OEIS|id=A053250}}.

:<math>
\psi(q) = \sum_{n > 0} \frac{q^{n^2}}{(q;q^2)_n} = \frac{q}{\prod_{n>0}(1-q^{4n})}\sum_{n\in \mathbf{Z}}\frac{(-1)^nq^{6n(n+1)}}{1-q^{4n+1}}
</math>  {{OEIS|id=A053251}}.

:<math>
\chi(q) = \sum_{n\ge 0} \frac{q^{n^2}}{\prod_{1\le i\le n}(1-q^i+q^{2i})} = \frac{1}{2 \prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^n(1+q^n)q^{n(3n+1)/2}}{1-q^n+q^{2n}}
</math>  {{OEIS|id=A053252}}.

:<math>
\omega(q) = \sum_{n\ge 0} \frac{q^{2n(n+1)}}{(q;q^2)^2_{n+1}} = \frac{1}{\prod_{n>0}(1-q^{2n})}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)} \frac{1+q^{2n+1}}{1-q^{2n+1}}}
</math>  {{OEIS|id=A053253}}.

:<math>
\nu(q) = \sum_{n\ge 0} \frac{q^{n(n+1)}}{(-q;q^2)_{n+1}} = \frac{1}{\prod_{n>0}(1-q^n)}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)/2}\frac{1-q^{2n+1}}{1+q^{2n+1}}}
</math>  {{OEIS|id=A053254}}.

:<math>
\rho(q) = \sum_{n\ge 0} \frac{q^{2n(n+1)}}{\prod_{0\le i\le n}(1+q^{2i+1}+q^{4i+2})} = \frac{1}{\prod_{n>0}(1-q^{2n})}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)} \frac{1-q^{4n+2}}{1+q^{2n+1}+q^{4n+2}}}
</math>  {{OEIS|id=A053255}}.

The first four 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):
:<math>\begin{align}
                                                       2\phi(-q) - f(q) &= f(q) + 4\psi(-q) = \theta_4(0,q)\prod_{r > 0}\left(1 + q^r\right)^{-1} \\
                                                        4\chi(q) - f(q) &= 3\theta_4^2(0, q^3)\prod_{r > 0}\left(1 - q^r\right)^{-1} \\
                                                   2\rho(q) + \omega(q) &= 3\left(\frac{1}{2}q^{-\frac{3}{8}}\theta_2(0, q^\frac{3}{2})\right)^2\prod_{r > 0}\left(1 - q^{2r}\right)^{-1} \\
                                   \nu(\pm q) \pm q\omega\left(q^2\right) &= \frac{1}{2}q^{-\frac{1}{4}}\theta_2(0, q)\prod_{r > 0}\left(1 + q^{2r}\right) \\
  f\left(q^8\right) \pm 2q\omega(\pm q) \pm 2q^3\omega\left(-q^4\right) &= \theta_3(0, \pm q)\theta_3^2\left(0, q^2\right)\prod_{r > 0}\left(1 - q^{4r}\right)^{-2} \\
                                      f(q^8) + q\omega(q) - q\omega(-q) &= \theta_3(0, q^4)
                \theta_3^2(0, q^2)\prod_{r > 0}\left(1 - q^{4r}\right)^{-2}
\end{align}</math>

===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 Watson.{{sfn|Watson|1937}} In his lost notebook he stated some further identities relating these functions, equivalent to the '''mock theta conjectures''',{{sfn|Andrews|Garvan|1989}} that were proved by Hickerson.{{sfn|Hickerson|1988a}} Andrews{{sfn|Andrews|1986}} found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight {{sfrac|1|2}}.
:<math>f_0(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q;q)_{n}}</math> {{OEIS|id=A053256}}
:<math>f_1(q) = \sum_{n\ge 0} \frac{q^{n^2+n}}{(-q;q)_{n}}</math> {{OEIS|id=A053257}}
:<math>\phi_0(q) = \sum_{n\ge 0} {q^{n^2}(-q;q^2)_{n}}</math> {{OEIS|id=A053258}}
:<math>\phi_1(q) = \sum_{n\ge 0} {q^{(n+1)^2}(-q;q^2)_{n}}</math> {{OEIS|id=A053259}}
:<math>\psi_0(q) = \sum_{n\ge 0} {q^{(n+1)(n+2)/2}(-q;q)_{n}}</math> {{OEIS|id=A053260}}
:<math>\psi_1(q) = \sum_{n\ge 0} {q^{n(n+1)/2}(-q;q)_{n}}</math> {{OEIS|id=A053261}}
:<math>\chi_0(q) = \sum_{n\ge 0} \frac{q^{n}}{(q^{n+1};q)_{n}} = 2F_0(q)-\phi_0(-q)</math> {{OEIS|id=A053262}}
:<math>\chi_1(q) = \sum_{n\ge 0} \frac{q^{n}}{(q^{n+1};q)_{n+1}} = 2F_1(q)+q^{-1}\phi_1(-q)</math> {{OEIS|id=A053263}}
:<math>F_0(q) = \sum_{n\ge 0} \frac{q^{2n^2}}{(q;q^2)_{n}}</math> {{OEIS|id=A053264}}
:<math>F_1(q) = \sum_{n\ge 0} \frac{q^{2n^2+2n}}{(q;q^2)_{n+1}}</math> {{OEIS|id=A053265}}
:<math>\Psi_0(q) =  -1 + \sum_{n \ge 0} \frac{ q^{5n^2}}{(1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^{5n-1})(1-q^{5n+1})}</math> {{OEIS|id=A053266}}
:<math>\Psi_1(q) = -1 + \sum_{n \ge 0} \frac{ q^{5n^2}}{(1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^{5n-2})(1-q^{5n+2}) }</math> {{OEIS|id=A053267}}

===Order 6===
Ramanujan{{sfn|Ramanujan|1988}} wrote down seven mock theta functions of order 6 in his lost notebook, and stated 11 identities between them, which were proved by Andrews and Hickerson.{{sfn|Andrews|Hickerson|1991}} 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 express
the remaining five sixth-order mock theta functions in terms of ''φ'' and ''ψ''. Berndt and Chan{{sfn|Berndt|Chan|2007}} discovered two more sixth-order functions.

The order 6 mock theta functions are:
:<math>\phi(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q;q)_{2n}}</math> {{OEIS|id=A053268}}
:<math>\psi(q) = \sum_{n\ge 0} \frac{(-1)^nq^{(n+1)^2}(q;q^2)_n}{(-q;q)_{2n+1}}</math> {{OEIS|id=A053269}}
:<math>\rho(q) = \sum_{n\ge 0} \frac{q^{n(n+1)/2}(-q;q)_n}{(q;q^2)_{n+1}}</math> {{OEIS|id=A053270}}
:<math>\sigma(q) = \sum_{n\ge 0} \frac{q^{(n+1)(n+2)/2}(-q;q)_n}{(q;q^2)_{n+1}}</math> {{OEIS|id=A053271}}
:<math>\lambda(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n}(q;q^2)_n}{(-q;q)_{n}}</math> {{OEIS|id=A053272}}
:<math>2\mu(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n+1}(1+q^n)(q;q^2)_n}{(-q;q)_{n+1}}</math> {{OEIS|id=A053273}}
:<math>\gamma(q) = \sum_{n\ge 0} \frac{q^{n^2}(q;q)_n}{(q^3;q^3)_{n}}</math> {{OEIS|id=A053274}}
:<math>\phi_{-}(q) = \sum_{n\ge 1} \frac{q^{n}(-q;q)_{2n-1}}{(q;q^2)_{n}}</math> {{OEIS|id=A153251}}
:<math>\psi_{-}(q) = \sum_{n\ge 1} \frac{q^{n}(-q;q)_{2n-2}}{(q;q^2)_{n}}</math> {{OEIS|id=A153252}}

===Order 7===
Ramanujan gave three mock theta functions of order 7 in his 1920 letter to Hardy. They were studied by Selberg,{{sfn|Selberg|1938}} who found asymptotic expansion for their coefficients, and by Andrews.{{sfn|Andrews|1986}} Hickerson{{sfn|Hickerson|1988b}} found representations of many of these functions as the quotients of indefinite theta series by modular forms of weight {{sfrac|1|2}}. Zwegers{{sfn|Zwegers|2001}}{{sfn|Zwegers|2002}} described their modular transformation properties.
* <math>\displaystyle F_0(q) = \sum_{n\ge 0}\frac{q^{n^2}}{(q^{n+1};q)_n}</math> {{OEIS|id=A053275}}
* <math>\displaystyle F_1(q) = \sum_{n\ge 0}\frac{q^{n^2}}{(q^{n};q)_n}</math> {{OEIS|id=A053276}}
* <math>\displaystyle F_2(q) = \sum_{n\ge 0}\frac{q^{n(n+1)}}{(q^{n+1};q)_{n+1}}</math> {{OEIS|id=A053277}}

These three mock theta functions have different shadows, so unlike the case of Ramanujan's order-3 and order-5 functions, there are no linear relations between them and ordinary modular forms.
The corresponding weak Maass forms are
:<math>
\begin{align}
M_1(\tau) & = q^{-1/168}F_1(q) + R_{7,1}(\tau) \\[4pt]
M_2(\tau) & = -q^{-25/168}F_2(q) + R_{7,2}(\tau) \\[4pt]
M_3(\tau) & = q^{47/168}F_3(q) + R_{7,3}(\tau)
\end{align}
</math>

where
:<math>R_{p,j}(\tau) = \!\!\!\! \sum_{n\equiv j\bmod p}\binom{12}{n}\sgn(n)\beta\left(\frac{n^2y}{6p}\right)q^{-n^2/24p}</math>

and
:<math>\beta(x) = \int_x^\infty u^{-1/2}e^{-\pi u} \, du</math>

is more or less the complementary [[error function]].
Under the metaplectic group, these three functions transform according to a certain 3-dimensional representation of the metaplectic group as follows

:<math>
\begin{align}
M_j\left(-\frac{1}{\tau}\right) & = \sqrt\frac{\tau}{7i}\,\sum_{k=1}^32\sin\left(\frac{6\pi jk}{7}\right)M_k(\tau) \\[6pt]
M_1(\tau+1) & = e^{-2\pi i/168} M_1(\tau), \\[6pt]
M_2(\tau+1) & = e^{-2\times 25\pi i/168} M_2(\tau), \\[6pt]
M_3(\tau+1) & = e^{-2\times 121\pi i/168} M_3(\tau).
\end{align}
</math>

In other words, they are the components of a level 1 vector-valued harmonic weak Maass form of weight&nbsp;{{sfrac|1|2}}.

===Order 8===
Gordon and McIntosh{{sfn|Gordon|McIntosh|2000}} found eight mock theta functions of order 8. They found five linear relations involving them, and expressed four of the functions as Appell–Lerch sums, and described their transformations under the modular group.
The two functions ''V''<sub>1</sub> and ''U''<sub>0</sub> were found earlier by Ramanujan{{sfn|Ramanujan|1988|loc=p. 8, eqn 1; p. 29 eqn 6}} in his lost notebook.
:<math>S_0(q) = \sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (-q^2;q^2)_n}</math>  {{OEIS|id=A153148}}
:<math>S_1(q) = \sum_{n\ge 0} \frac{q^{n(n+2)} (-q;q^2)_n }{ (-q^2;q^2)_n}</math>  {{OEIS|id=A153149}}
:<math>T_0(q) = \sum_{n\ge 0} \frac{q^{(n+1)(n+2)} (-q^2;q^2)_n }{ (-q;q^2)_{n+1}}</math>  {{OEIS|id=A153155}}
:<math>T_1(q) = \sum_{n\ge 0} \frac{q^{n(n+1)} (-q^2;q^2)_n }{ (-q;q^2)_{n+1}}</math> {{OEIS|id=A153156}}
:<math>U_0(q) = \sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (-q^4;q^4)_n}</math>  {{OEIS|id=A153172}}
:<math>U_1(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2} (-q;q^2)_n }{ (-q^2;q^4)_{n+1}}</math> {{OEIS|id=A153174}}
:<math>V_0(q) = -1+2\sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (q;q^2)_n} = -1+2\sum_{n\ge 0} \frac{q^{2n^2} (-q^2;q^4)_n}{(q;q^2)_{2n+1}}</math>  {{OEIS|id=A153176}}
:<math>V_1(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2} (-q;q^2)_n }{ (q;q^2)_{n+1}} = \sum_{n\ge 0} \frac{q^{2n^2+2n+1} (-q^4;q^4)_n}{(q;q^2)_{2n+2}}</math>  {{OEIS|id=A153178}}

===Order 10===
Ramanujan{{sfn|Ramanujan|1988|p=9}} listed four order-10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by Choi.{{sfn|Choi|1999}}{{sfn|Choi|2000}}{{sfn|Choi|2002}}{{sfn|Choi|2007}}
* <math> \phi(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(q;q^2)_{n+1}}</math> {{OEIS|id=A053281}}
* <math> \psi(q)=\sum_{n\ge 0}\frac{q^{(n+1)(n+2)/2}}{(q;q^2)_{n+1}}</math> {{OEIS|id=A053282}}
* <math> \Chi(q)=\sum_{n\ge 0}\frac{(-1)^nq^{n^2}}{(-q;q)_{2n}}</math> {{OEIS|id=A053283}}
* <math> \chi(q)=\sum_{n\ge 0}\frac{(-1)^nq^{(n+1)^2}}{(-q;q)_{2n+1}}</math> {{OEIS|id=A053284}}

==Notes==
{{reflist}}

==References==
{{refbegin|30em}}
*{{Citation| title = On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
 | last = Andrews | first = George E. | year = 1966
 | journal = [[American Journal of Mathematics]]
 | volume = 88 | issue = 2 | pages = 454–490
 | doi = 10.2307/2373202 | issn = 0002-9327 | jstor = 2373202 | mr = 0200258
}}
*{{Citation| title = The fifth and seventh order mock theta functions
 | last = Andrews | first = George E. | year = 1986
 | journal = [[Transactions of the American Mathematical Society]]
 | volume = 293 | issue = 1 | pages = 113–134
 | doi = 10.2307/2000275 | issn = 0002-9947 | jstor = 2000275 | mr = 814916
 | doi-access = free
}}
*{{Citation| chapter = Ramanujan's fifth order mock theta functions as constant terms
 | last = Andrews | first = George E. | year = 1988
 | title = Ramanujan revisited (Urbana-Champaign, Ill., 1987)
 | publisher = [[Academic Press]] | location = Boston, MA
 | pages = 47–56
 | mr = 938959
}}
*{{Citation| chapter = Mock theta functions
 | last = Andrews | first = George E. | year = 1989
 | title = Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987)
 | publisher = [[American Mathematical Society]] | location = Providence, R.I.
 | volume = 49 | series = Proc. Sympos. Pure Math.
 | pages = 283–298
 | mr = 1013178
}}
*{{Citation| title = Ramanujan's lost notebook. VI. The mock theta conjectures
 | last1 = Andrews | first1 = George E.
 | last2 = Garvan | first2 = F. G.
 | journal = [[Advances in Mathematics]]
 | year = 1989 | volume = 73 | issue = 2 | pages = 242–255
 | doi = 10.1016/0001-8708(89)90070-4 | issn = 0001-8708 | mr = 987276
 | doi-access = free
}}
*{{Citation| title = Ramanujan's lost notebook. VII. The sixth order mock theta functions
 | last1 = Andrews | first1 = George E.
 | last2 = Hickerson | first2 = Dean
 | journal = [[Advances in Mathematics]]
 | year = 1991 | volume = 89 | issue = 1 | pages = 60–105
 | doi = 10.1016/0001-8708(91)90083-J | issn = 0001-8708 | mr = 1123099
 | doi-access = free
}}
*{{citation| title = Sur les fonctions doublement périodiques de troisième espèce
 | last = Appell | first = P. | year = 1884
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | volume = 1 | pages = 135–164
 | doi = 10.24033/asens.236
 | doi-access = free
}}
*{{Citation| title = Sixth order mock theta functions
 | last1 = Berndt | first1 = Bruce C.
 | last2 = Chan | first2 = Song Heng
 | journal = [[Advances in Mathematics]]
 | year = 2007 | volume = 216 | issue = 2 | pages = 771–786
 | doi = 10.1016/j.aim.2007.06.004 | issn = 0001-8708 | mr = 2351377
 | doi-access = free
}}
*{{Cite journal | title = q-series and weight 3/2 Maass forms
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Folsom | first2 = Amanda
 | last3 = Ono | first3 = Ken
 | author1-link = Kathrin Bringmann
 | author2-link = Amanda Folsom
 | author3-link = Ken Ono
 | journal = Compositio Mathematica
 | year = 2009 | volume = 145 | issue = 3 | pages = 541–552
 | url = https://afolsom.people.amherst.edu/BringmannFolsomOno-Compositio.pdf
 | doi = 10.1112/S0010437X09004072
| s2cid = 7688222 | doi-access = free
 }}
*{{Citation| title = The f(q) mock theta function conjecture and partition ranks
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Inventiones Mathematicae]]
 | year = 2006 | volume = 165 | issue = 2 | pages = 243–266
 | url = https://uva.theopenscholar.com/files/ken-ono/files/098_8.pdf
 | bibcode = 2006InMat.165..243B | doi = 10.1007/s00222-005-0493-5 | doi-access = free | issn = 0020-9910 | mr = 2231957
| s2cid = 120388256 }}
*{{Citation| title = Lifting cusp forms to Maass forms with an application to partitions
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]]
 | year = 2007 | volume = 104 | issue = 10 | pages = 3725–3731
 | bibcode = 2007PNAS..104.3725B | doi = 10.1073/pnas.0611414104 | doi-access = free | issn = 0027-8424 | mr = 2301875 | pmc = 1820651 | pmid = 17360420
}}
*{{Citation| title = Dyson's ranks and Maass forms
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Annals of Mathematics]]
 | year = 2010 | volume = 171 | page = 419
 | url = https://uva.theopenscholar.com/files/ken-ono/files/100_8.pdf
 | doi = 10.4007/annals.2010.171.419
 | doi-access = free
 }}
*{{Citation| title = On two geometric theta lifts
 | last1 = Bruinier | first1 = Jan Hendrik
 | last2 = Funke | first2 = Jens
 | journal = [[Duke Mathematical Journal]]
 | year = 2004 | volume = 125 | issue = 1 | pages = 45–90
 | arxiv = math/0212286 | doi = 10.1215/S0012-7094-04-12513-8 | issn = 0012-7094 | mr = 2097357
| s2cid = 2078210 }}
*{{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook
 | last = Choi | first = Youn-Seo | year = 1999
 | journal = Inventiones Mathematicae
 | volume = 136 | issue = 3 | pages = 497–569
 | bibcode = 1999InMat.136..497C | doi = 10.1007/s002220050318 | doi-access = | issn = 0020-9910 | mr = 1695205
| s2cid = 125193659 }}
*{{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. II
 | last = Choi | first = Youn-Seo | year = 2000
 | journal = [[Advances in Mathematics]]
 | volume = 156 | issue = 2 | pages = 180–285
 | doi = 10.1006/aima.2000.1948 | issn = 0001-8708 | mr = 1808245
 | doi-access = free
}}
*{{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. IV
 | last = Choi | first = Youn-Seo | year = 2002
 | journal = Transactions of the American Mathematical Society
 | volume = 354 | issue = 2 | pages = 705–733
 | doi = 10.1090/S0002-9947-01-02861-6 | issn = 0002-9947 | jstor = 2693766 | mr = 1862564
 | doi-access = free
}}
*{{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. III
 | last = Choi | first = Youn-Seo | year = 2007
 | journal = Proceedings of the London Mathematical Society
 | volume = 94 | issue = 1 | pages = 26–52
 | doi = 10.1112/plms/pdl006 | issn = 0024-6115 | mr = 2293464
| s2cid = 122732998 }}
*{{cite arXiv| mode = cs2| title = Quantum Black Holes, Wall Crossing, and Mock Modular Forms
 | last1 = Dabholkar | first1 = Atish
 | last2 = Murthy | first2 = Sameer
 | last3 = Zagier | first3 = Don
 | year = 2012
 | class = hep-th
 | eprint = 1208.4074
}}
*{{Citation| title = Some asymptotic formulae for the mock theta series of Ramanujan
 | last = Dragonette | first = Leila A. | year = 1952
 | journal = Transactions of the American Mathematical Society
 | volume = 72 | issue = 3 | pages = 474–500
 | doi = 10.2307/1990714 | issn = 0002-9947 | jstor = 1990714 | mr = 0049927
 | doi-access = free
}}
*{{Citation| title = Basic hypergeometric series and applications
 | last = Fine | first = Nathan J. | year = 1988
 | publisher = American Mathematical Society | location = Providence, R.I.
 | volume = 27 | series = Mathematical Surveys and Monographs
 | isbn = 978-0-8218-1524-3 | mr = 956465
}}
*{{Citation| title = The coefficients of the ω(q) mock theta function
 | last = Garthwaite | first = Sharon Anne | year = 2008
 | journal = International Journal of Number Theory
 | volume = 4 | issue = 6 | pages = 1027–1042
 | doi = 10.1142/S1793042108001869 | issn = 1793-0421 | mr = 2483310
}}
*{{Citation| title = Some eighth order mock theta functions
 | last1 = Gordon | first1 = Basil
 | last2 = McIntosh | first2 = Richard J.
 | author1-link = Basil Gordon
 | journal = Journal of the London Mathematical Society
 | year = 2000 | volume = 62 | issue = 2 | pages = 321–335
 | url = http://journals.cambridge.org/action/displayAbstract?aid=63263
 | doi = 10.1112/S0024610700008735 | issn = 0024-6107 | mr = 1783627
| s2cid = 123182909 | url-access = subscription}}
*{{Citation| title = A proof of the mock theta conjectures
 | last = Hickerson | first = Dean | year = 1988a
 | journal = Inventiones Mathematicae
 | volume = 94 | issue = 3 | pages = 639–660
 | bibcode = 1988InMat..94..639H | doi = 10.1007/BF01394279 | doi-access = | issn = 0020-9910 | mr = 969247
| s2cid = 122492320 }}
*{{Citation| title = On the seventh order mock theta functions
 | last = Hickerson | first = Dean | year = 1988b
 | journal = Inventiones Mathematicae
 | volume = 94 | issue = 3 | pages = 661–677
 | bibcode = 1988InMat..94..661H | doi = 10.1007/BF01394280 | doi-access = | issn = 0020-9910 | mr = 969247
| s2cid = 121384412 }}
*{{Citation| title = Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
 | last1 = Hirzebruch | first1 = Friedrich
 | last2 = Zagier | first2 = Don
 | author1-link = Friedrich Hirzebruch
 | journal = Inventiones Mathematicae
 | year = 1976 | volume = 36 | pages = 57–113
 | bibcode = 1976InMat..36...57H | doi = 10.1007/BF01390005 | hdl = 21.11116/0000-0004-399B-E | issn = 0020-9910 | mr = 0453649
 | s2cid = 56568473 | hdl-access = free
}}
*{{Citation| title = Modular forms and quantum invariants of 3-manifolds
 | last1 = Lawrence | first1 = Ruth
 | last2 = Zagier | first2 = Don
 | journal = The Asian Journal of Mathematics
 | year = 1999 | volume = 3 | issue = 1 | pages = 93–107
 | doi = 10.4310/AJM.1999.v3.n1.a5 | issn = 1093-6106 | mr = 1701924
 | doi-access = free
}}
*{{citation| title = Bemerkungen zur Theorie der elliptischen Funktionen
 | last = Lerch | first = M. | year = 1892
 | journal = Jahrbuch über die Fortschritte der Mathematik
 | volume = 24 | pages = 442–445
}}
*{{Citation| title = Second order mock theta functions
 | last = McIntosh | first = Richard J. | year = 2007
 | journal = [[Canadian Mathematical Bulletin]]
 | volume = 50 | issue = 2 | pages = 284–290
 | doi = 10.4153/CMB-2007-028-9 | doi-access=free | issn = 0008-4395 | mr = 2317449
| s2cid = 119499438 }}
*{{Citation| title = The lost notebook and other unpublished papers
 | last = Ramanujan | first = Srinivasa | year = 1988
 | publisher = [[Springer-Verlag]] | location = Berlin, New York
 | isbn = 978-3-540-18726-4 | mr = 947735
}}
*{{Citation| title = Collected papers of Srinivasa Ramanujan
 | last = Ramanujan | first = Srinivasa | year = 2000
 | location = Providence, R.I.
 | volume = 123 | issue = 3104 | page = 631
 | journal = Nature
 | bibcode = 1929Natur.123..631L | doi = 10.1038/123631a0 | isbn = 978-0-8218-2076-6 | mr = 2280843
| s2cid = 44812911 }}
*{{citation| title = Über die Mock-Thetafunktionen siebenter Ordnung. (On the mock theta functions of seventh order)
 | last = Selberg | first = A. | year = 1938
 | journal = Archiv for Mathematik og Naturvidenskab
 | volume = 41 | pages = 3–15
}}, reprinted in volume I of his collected works
*{{Citation| title = Higher-level Appell functions, modular transformations, and characters
 | last1 = Semikhatov | first1 = A. M.
 | last2 = Taormina | first2 = A.
 | last3 = Tipunin | first3 = I. Yu
 | author2-link = Anne Taormina
 | journal = Communications in Mathematical Physics
 | year = 2005 | volume = 255 | issue = 2 | pages = 469–512
 | arxiv = math/0311314 | bibcode = 2005CMaPh.255..469S | doi = 10.1007/s00220-004-1280-7 | issn = 0010-3616 | mr = 2129953
| s2cid = 14466569 }}
*{{Citation| chapter = Modular forms of weight 1/2
 | last1 = Serre | first1 = Jean-Pierre
 | last2 = Stark | first2 = H. M.
 | author1-link = Jean-Pierre Serre
 | year = 1977
 | title = Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976)
 | publisher = [[Springer-Verlag]] | location = Berlin, New York
 | volume = 627 | series = Lecture Notes in Mathematics
 | pages = 27–67
 | doi = 10.1007/BFb0065296 | isbn = 978-3-540-08530-0 | mr = 0472707
}}
*{{Citation| title = The Non-Compact Elliptic Genus: Mock or Modular
 | last = Troost | first = J. | year = 2010
 | journal = Journal of High Energy Physics
 | volume = 2010 | issue = 6 | page = 104
 | arxiv = 1004.3649 | bibcode = 2010JHEP...06..104T | doi = 10.1007/JHEP06(2010)104
| s2cid = 84838021 }}
*{{citation| title = The Final Problem: An Account of the Mock Theta Functions
 | last = Watson | first = G. N. | year = 1936
 | journal = Journal of the London Mathematical Society
 | volume = 11 | pages = 55–80
 | doi = 10.1112/jlms/s1-11.1.55
}}
*{{citation| title = The Mock Theta Functions (2)
 | last = Watson | first = G. N. | year = 1937
 | journal = Proceedings of the London Mathematical Society
 | volume = s2-42 | pages = 274–304
 | doi = 10.1112/plms/s2-42.1.274
}}
*{{Citation| title = Nombres de classes et formes modulaires de poids 3/2
 | last = Zagier | first = Don | year = 1975
 | journal = Comptes Rendus de l'Académie des Sciences, Série A et B
 | volume = 281 | issue = 21 | pages = Ai, A883–A886
 | issn = 0151-0509 | mr = 0429750
}}
*{{Citation| title = Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann)
 | last = Zagier | first = Don | year = 2009
 | orig-year = First published 2007
 | volume = 326 | pages = 143–164
 | department = Séminaire Bourbaki. Exp. 986
 | journal = Astérisque
 | url = http://people.mpim-bonn.mpg.de/zagier/files/aster/326/fulltext.pdf
 | issn = 0303-1179 | mr = 2605321
 | ref = {{harvid|Zagier|2007}}
}}
*{{Citation| chapter = Mock θ-functions and real analytic modular forms
 | last = Zwegers | first = S. P. | year = 2001
 | title = q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000)
 | publisher = American Mathematical Society | location = Providence, R.I.
 | volume = 291 | series = Contemp. Math.
 | pages = 269–277
 | isbn = 978-0-8218-2746-8 | mr = 1874536
}}
*{{Citation| title = Mock Theta Functions
 | last = Zwegers | first = S. P. | year = 2002
 | publisher = Utrecht PhD thesis
 | url = http://igitur-archive.library.uu.nl/dissertations/2003-0127-094324/inhoud.htm
 | isbn = 90-393-3155-3
}}
{{refend}}

==Further reading==
{{refbegin}}
*{{Citation| chapter = Mock theta functions, ranks and Maass forms
 | last = Ono | first = Ken | year = 2008
 | author-link = Ken Ono
 | title = Surveys in Number Theory
 | editor-last = Alladi | editor-first = Krishnaswami | editor-link = Krishnaswami Alladi
 | publisher = [[Springer-Verlag]]
 | volume = 17 | series = Developments in Mathematics
 | pages = 119–141
 | isbn = 978-0-387-78509-7 | zbl = 1183.11064
 | ref = none
}}
{{refend}}

==External links==
* [https://web.archive.org/web/20081221143605/http://www.mpim-bonn.mpg.de/Events/This+Year+and+Prospect/Mock+theta+functions/ International Conference: Mock theta functions and applications 2009]
* [https://web.archive.org/web/20081204111537/http://www.math.psu.edu/andrews/biblio.html Papers on mock theta functions] by [[George Andrews (mathematician)|George Andrews]]
* [http://www.mi.uni-koeln.de/~kbringma/papers.html Papers on mock theta functions] by [[Kathrin Bringmann]]
* [https://web.archive.org/web/20100620082612/http://www.math.wisc.edu/~ono/reprints/index.html Papers on mock theta functions] by [[Ken Ono]]
* [https://web.archive.org/web/20081023234404/http://mathsci.ucd.ie/~zwegers/ Papers on mock theta functions] by [[Sander Zwegers]]
* {{mathworld|urlname=MockThetaFunction|title=Mock Theta Function}}

[[Category:Modular forms]]
[[Category:Q-analogs]]
[[Category:Srinivasa Ramanujan]]