As Euler showed the harmonic series 1 + 1/2 + 1/3 + 1/4 +..... + n is approximated as log n + λ (for a finite value of n).
And as λ (the Euler-Masceroni constant) is a constant = .5772 approx this means that for large n, log n is approximated by the harmonic series.
This means therefore that perhaps the simplest expression for the frequency of prime distribution is given as the sum of the harmonic series for n terms divided by n (which becomes increasingly accurate for larger n).
Looked at another way the sum of the harmonic series (for large n) approximates the average spread or gap as between prime numbers in the region of n.
Therefore as the sum of the first million terms for example of the harmonic series = 14.384 (approx). Therefore the average gap as between primes in the region of 1,000,000 is roughly 14. Though this approximation is not yet very accurate, the approximation would greatly improve (in relative terms) as the value of n increases.
Now it is well known that the average spread as between primes continually increases as the value of n increases.
What I find particularly striking in this regard is that the increase in the average spread (or gap) as between primes as we increase n by 1 is given by 1/n.
So for example as we increase n 1,000,000 to 1,000,001 the increase in the average gap as between primes is 1/1,000,000.
(More accurately as we increase n from 999,999.5 to 1,000,000.5 the average gap between primes increases by 1/1,000,000).
This result can easily be demonstrated through differentiation of log n + λ (with respect to n) which results in 1/n.
Now if we multiply the simple expression for the general frequency of primes i.e. n/log n by 1/n we obtain 1/log n (which represents the probability that n is prime).
Thus, we can say that the product of the general frequency of prime distribution and the change in the average gap as between primes (for large n) approximates well the probability that n is prime.
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