# 3rd order mock theta functions

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## introduction

• Ramanujan's 3rd order mock theta function is defined by

$f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}$

• the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by of (Andrews 1966) and Dragonette (1952) converges to the coefficients (Bringmann & Ono 2006).
• In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.

## asymptotic behavior at roots of unity

• the series converges for $$|q|<1$$ and the roots of unity $$q$$ at odd order
• For even order roots of unity, $$f(q)$$ has exponential singularities but there is a nice result to describe this behavior
• let us define $b(q)=(1-q)(1-q^3)(1-q^5)\cdots (1-2q+2q^4-\cdots)$, or we can write it as $b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)$
• let $$\zeta$$ be even $$2k$$ order root of unity

$\lim_{q\to \zeta} f(q)-(-1)^k b(q)=-4\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}$

• if $$k=2$$, as $$q\to i$$, $$f(q)-b(q)\to 4i$$

## harmonic weak Maass form

• We have a weight k=1/2, harmonic weak Maass form $$h_3$$ under $$\Gamma(2)$$ defined by $h_3(\tau)=q^{-1/24}f(q)+R_3(q)$ where

$R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}$ where $\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt$

• Note that this can be rewritten as $R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz$

where $$g$$ is the shadow $g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}$

• shadow = weight 3/2 theta function
• $$\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots$$
• $$M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz$$