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Thursday, September 17, 2015

**Abstract:** The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdős–Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We show that the Erdős–Turán Law can be extended to random permutations chosen according to the so-called *generalized Ewens measure* and to a *generalized weighted measure* with polynomially growing cycle weights. Furthermore, we establish for the *generalized Ewens measure* a local limit theorem as well as, under some extra moment condition, a precise large deviation estimate and also show that the expectation of the logarithm of the order has a remarkable connection with the Riemann hypothesis. In addition, we provide a precise large deviation estimate for random permutations with polynomial growing cycle weights.