We study the signature $\sigma_g(\frac q p)$ of $\mathrm{ SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $\zeta=e^{ i\pi q/p}$. We prove that $\frac{ 1}{ p^2}\sigma_2(\frac{ q}{ p})$ converges to $\Lambda(\theta)=\frac{ 16}{ \pi^3}\sum\limits_{ n\ge 1, \textrm{ odd}}\frac{ 1}{ n^3\sin(n\pi\theta)}$ when $\frac{ q}{ p}$ goes to an irrational number $\theta\in [0,1]$ under certain conditions. We also observe that the function $\Lambda(\theta)$ is the boundary value of an Eichler integral of a level $2$ modular form of weight $4$, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums.
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