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Using the multiple cover formula of Y. Toda for counting invariants ofsemistable twisted sheaves over twisted local K3 surfaces we calculate the$\SU(r)/\zz_r$-Vafa-Witten invariants for K3 surfaces for any rank $r$ for theLanglands dual group $\SU(r)/\zz_r$ of the gauge group $\SU(r)$. We generalizeand prove the S-duality conjecture of Vafa-Witten for K3 surfaces in any rank$r$ based on the result of Tanaka-Thomas for the $\SU(r)$-Vafa-Witteninvariants.

A proof of all ranks S-duality conjecture for K3 surfaces