Functional Analysis and Additive Arithmetic Functions

1. A function is arithmetic if it is defined on the positive integers. Those arithmetic functions which assume real values and satisfy f(ab)-f(a)+f(b) for mutually prime integers a, b are classically called additive. The following examples illustrate the interest of these functions, both for themselves and for their applications. An additive function is defined by its values on the prime powers. I shall denote a typical prime power by q, and the prime of which it is a power by q0. A well-known additive function is o)(n) which, with o)(q) = 1, counts the number of distinct prime divisors of n. Let vx(n\ . . .) denote the frequency amongst the positive integers not exceeding x of those for which property ... holds. Then as x -» oo