Variance of a strongly additive function defined on random permutations

Abstract

Inspired by unfading popularity of the Turán–Kubilius inequality for additive number theoretic functions within the last decades, we examine the variance of additive functions defined on random permutations uniformly taken from the symmetric group. Extending the optimal estimate achieved in 2018 by Klimavičius and Manstavičius for the case of completely additive functions, we obtain asymptotically sharp upper and lower bounds when the functions are strongly additive. The upper estimates are analogous to that established in number theory by Kubilius in 1985.