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We consider three different families of Vafa-Witten invariants of $K3$surfaces. In each case, the partition function that encodes the Vafa-Witteninvariants is given by combinations of twisted Dedekind $\eta$-functions. Byutilising known properties of these $\eta$-functions, we obtain exact formulaefor each of the invariants and prove that they asymptotically satisfy allhigher-order Tur\'{a}n inequalities.

Exact formulae and Tur\'{a}n inequalities for Vafa-Witten invariants of $K3$ surfaces