# Quantum modular forms

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## Kontsevich's strange function

• definition

$F(q)=\sum_{n=0}^{\infty}(q)_n$

• originated from quantum invariants of trefoil knot
• if $$F(x)=F(e^{2\pi i x})$$, then

$\zeta_{24k}^{-1}F(\frac{-1}{k})\sim \sqrt{-i}k^{3/2}\zeta_{24}^kF(k)+g(k)$

• theorem (Zagier)

Let $\phi(x)=e^{\pi i x /12}F(e^{2\pi i x})$ $$\phi : \mathbb{Q} \to \mathbb{C}$$ satisfies $\phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x)$ where $$g:\mathbb{R}\to \mathbb{C}$$ is a $$C^{\infty}$$ function

• Strange identity

$F(q^{-1})=-\frac{1}{2}\sum_{n=1}^{\infty}n \left(\frac{12}{n}\right)q^{-\frac{n^2-1}{24}}$ with $$q=e^{2\pi i x}$$

• related to the partial theta function $$\tilde{\eta}(q)$$

## generating function of unimodal sequences

• generating function of unimodal sequences

$U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1}$

$R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}$

$C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}$

• limit formula $$\zeta_b=e^{2\pi i/b}$$, $$1\le a <b$$, for every root of unity $$\zeta$$, there exists an integer $$c$$ such that

$\lim_{q\to \zeta} R(\zeta_{b}^{a};q)-\zeta_{b^2}^{c} C(\zeta_{b}^{a};q)=-(1-\zeta_{b}^{a})(1-\zeta_{b}^{-a})U(\zeta_{b}^{a};\zeta)$

### special case

• If $$b=2$$ and $$a=1$$, then $$\zeta_{b}^{a}=-1$$
• $$U(-1;\zeta)$$ becomes a finite sum if $$\zeta$$ is a root of unity

$U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}$

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

### Kontsevich's strange function

$U(q)=F(q^{-1})$

### non-holomorphic modular form

$q^{-1/24}U(q)+\int +\int$ is a non-holomorphic modular form of weight 3/2

## $$\sigma$$ and $$\sigma^{*}$$

• $$\sigma(q)=2\sum_{n=0}^{\infty}(-1)^n (q;q)_n$$
• (Cohen) $$\sigma(q)=-\sigma^{*}(q^{-1})$$ for every root of unity
• let $$f(x)=q^{1/24}\sigma(q)$$ where $$q=e^{2\pi i x}$$
• (Lewis-Zagier) $$f : \mathbb{Q} \to \mathbb{C}$$ satisfies

$\frac{1}{2x+1}f(\frac{x}{2x+1})=e^{\pi i/12}f(x)+h(x)$ where $$h$$ is $$C^{\infty}$$ on $$\mathbb{R}$$ and real analytic except at $$x=-1/2$$