On Large Deviations of Additive Functions

Abstract. We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. The Kac-Kubilius model suggests that the distribution of values of a given additive function can be modeled by a sum of random variables. We show that the model is accurate when one is looking at values of the additive function around its mean, but fails, by a constant multiple, for large values of the additive function. We believe that this phenomenon arises, because the model breaks down for the values of the additive function on the “large” primes. In the second part of the paper, we are motivated by a question of Elliott, to understand how much the distribution of values of the additive function on primes determines, and is determined by, the distribution of values of the additive function on all of the integers. For example, our main theorem implies that a positive, strongly additive function is roughly Poisson distributed on the integers if and only if it is 1 + o(1) or o(1) on almost all primes.