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Abstract. The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a rate of convergence to the normal distribution for the central limit theorem. Some generalizations are provided.

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions P 1 and P 2 (GW(P 1 ) and GW(P 2 )) where P 1 and P 2 are supported on positive integers and P 1 dominates P 2 stochastically. We prove that the speed of the walk on GW(P 1 ) is bigger than the same on GW(P 2 ) when the bias is larger than a threshold depending on P 1 and P 2 . This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.2000 Mathematics Subject Classification. 60K37; 60J80; 60G50.

Abstract. Erdős-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations and moderate deviations for this problem in a very general setting for a wide class of additive functions.