On a conjecture by Erdos and its extension to additive functions on the set of pairs of integers

It was conjectured by P. Erdos in 1947, that if f is a real-valued additive arithmetic function and J is a bounded interval such that {m: f(m) is an element of J} has positive density, then f has a distribution. It was shown to be true under some additional assumptions, in late sixties and seventies. An extension of this to arithmetic functions on the set of pairs of positive integers is partially solved in this paper. The natural density on a set of pairs of positive integers can be defined in more than one way. The solution to the problem seems to depend on the particular density considered.