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We propose a general set of constraints on the partition function of quarterBPS dyons in any N=4 supersymmetric string theory by drawing insight from knownexamples, and study the consequences of this proposal. The main ingredients ofour analysis are duality symmetries, wall crossing formula and black holeentropy. We use our analysis to constrain the dyon partition function for twohitherto unknown cases -- the partition function of dyons of torsion two (i.e.gcd(Q\wedge P)=2) in heterotic string theory on T^6 and the partition functionof dyons carrying untwisted sector electric charge in Z_2 CHL model. With thehelp of these constraints we propose a candidate for the partition function ofdyons of torsion two in heterotic string theory on T^6. This leads to a novelwall crossing formula for decay of quarter BPS dyons into half BPS dyons withnon-primitive charge vectors. In an appropriate limit the proposed formulareproduces the known result for the spectrum of torsion two dyons in gaugetheory.

Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two