P. Erdös, using analytic
theorems, has proven the following results: Let f(x) be the number of integers
m such that ϕ(m) ≦ x, where ϕ is the Euler function, and let g(x) be the
number of integers n such that σ(n) ≦ x, where σ is the usual sum of divisors
function. Then there are positive (but undetermined) constants 01 and c2
such that f(x) = c1x + o(x) and g(x) = c2(x) + o(x). The constants c1 and
c2 can be calculated using complex analysis including the Wiener-Ikehara
Theorem. A major purpose of this paper is to give an elementary proof that
limx→∞f(x)∕x exists and, in the process, calculate the value of the limit.
These considerations of multiplicity motivate a generalization of natural
density which counts multiplicity. This paper contains an investigation of this
generalization.
