Numerical experiments on coefficients of instanton partition functions

We analyze the coefficients of partition functions of Vafa-Witten theory for the complex projective plane $\mathbb{CP}^2$. We experimentally study the growth of the coefficients for gauge group $SU(2)$ and $SU(3)$, which are examples of mock modular forms of depth $1$ and 2 respectively. We also introduce the notion of ``mock cusp form'', and study an example of weight 3 related to the $SU(3)$ partition function. Numerical experiments on the first 200 coefficients suggest that the coefficients of a mock modular form of weight $k$ grow as the coefficients of a modular form of weight $k$, that is to say as $n^{k-1}$. On the other hand the coefficients of the mock cusp form appear to grow as $n^{3/2}$, which exceeds the growth of classical cusp forms of weight 3. We provide bounds using saddle point analysis, which however largely exceed the experimental observation.