Sign of BPS index for ${\cal N}=4$ dyons

In this paper we argue how the sign changes on an average for the positive weight mock modular forms associated with the ${\cal N}=4$ type II string black holes compactified on orbifolds of $K3\times T^2$. The orbifolds of order $N$ act with $g'\in[M_{23}]$ an order $N$ symplectic orbifold on $K3$ and a $1/N$ shift in one of the circles of the torus $T^2$. We expand the inverse Siegel modular forms of subgroups of $Sp_2(\mathbb{Z})$ for the magnetic charge $P^2=2$ in terms of mock Jacobi forms and Appell Lerch sums. We analyze the average growth of the coefficients of these mock modular forms after theta decomposition and removing inverse eta products. In particular we remove the contribution of the fundamental string which rightfully dominates the growth of the positive weight modular forms after the first few coefficients and ensures the positivity of the helicity trace index $-B_6$. Using numerics and limits of divisor sum function we predict the sign of these mock modular forms. We also observe that the cusp forms associated with the non-geometric orbifolds of $K3$ can only contribute for sign changes up to the first few terms hence their contribution can be neglected for large electric charges.