Mean of highest exponent in prime factorization of all integers ≥ 2

Question

For any natural number $n > 1$, define $E(n)$,to be the highest exponent to which a prime divides it. For instance, $E(12)=E(36)=2$. Show that $$\lim_{N \to \infty} \frac{1}{N} \sum\limits_{n=2}^{N} E(n)$$ exists and find its value

Answer 1

Here are some suggestions to approximate the limit.

Consider F(n) to be the greater of 1 and the highest power of 2 that appears in the prime factorization of n. Show what the limit of the average over n of F(n) is as n gets large.

Let G_p(n) be defined like F(n), except replace 2 by an arbitrary prime p.

Note that F(n) <= E(n), and that E(n) = max(G_p(n) over all primes p), so that if limit of the average over n of E(n) exists, you can bound it using a sum of limits involving G_p.

For more accurate estimates, consider inclusion-exclusion.