arXiv:1109.0033v1 [math.NT] 31 Aug 2011 A STRUCTURE THEOREM IN PROBABILISTIC NUMBER THEORY MAKSYM RADZIWI L L 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. This is a consequence of a structure theorem making clear the inter-relation between the distribution of an additive function on the integers, and its distribution on the primes. 1. Introduction. Let g be an additive function (that is, g (mn)= g (m)+g (n) for (m,n) = 1 and g (p k )= g (p) on the primes). According to a probabilistic model of Mark Kac [6], the distribution of the g (n)’s (with n  x and x large) is predicted by the random variable, (1)  px g (p)X p . In (1) the X p ’s are independent random variables with P(X p = 1) = 1/p and P(X p = 0) = 1 − 1/p. According to the model, for most n  x the values g (n) cluster around the mean µ(g ; x) of (1), and within an error of O (σ(g ; x)). Here µ(g ; x) and σ 2 (g ; x) are respectively the mean and the variance of (1). Thus, µ(g ; x)=  px g (p) p and σ 2 (g ; x)=  px g (p) 2 p ·  1 − 1 p  . When looking at large values of g it is natural to consider (2) 1 x · #  n  x : g (n) − µ(g ; x) σ(g ; x)  Δ  , with Δ growing to infinity with x. For an additive function g (with, say, g (p)= O(1) and σ(g ; x) →∞) in the range Δ  o(σ 1/3 ) the frequency (2) is asymptotic to a normal distribution. For Δ  ε · σ 1/3 the distribution of (2) is no more Gaussian, and a rather complicated asymptotic formulae emerges (see Theorem 2 in [9] and [8] for a probabilistic analogue). Relatively little is known beyond the range Δ ≍ σ (except for ω(n), see [3], [4]). Our objective in this paper is to study the interralation between the distribution of large values of an additive function on the integers (that is, (2) with Δ growing to infinity) and the distribution of the values of the additive function on the primes. To fix ideas, and to simplify some of our arguments, we will restrict ourselves to the following class of aditive functions. Definition 1. An additive function g belongs to C if and only if • g is strongly additive and g (p)= O(1). 2010 Mathematics Subject Classification. Primary: 11N64, Secondary: 11N60, 11K65, 60F10. The author is partially supported by a NSERC PGS-D award. 1