Partition functions of torsion > 1 dyons in heterotic string theory on T6

The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T-6 is known to correctly produce the degeneracy of dyons of torsion 1, i.e. dyons for which gcd(Q Lambda P) = 1. We propose a generalization of this formula for dyons of arbitrary torsion. Our proposal satisfies the constraints coming from S-duality invariance, wall crossing formula, black hole entropy and the gauge theory limit. Furthermore using our proposal we derive a general wall crossing formula that is valid even when both the decay products are non-primitive half-BPS dyons.