Zoning in on the Primes

We have seen how important the harmonic series is with respect to the nature of prime numbers.

Once again - as an initial approximation - the sum of the first n terms of the series provides an estimate of the gap as between primes (in the region of n).
Secondly the reciprocal of n provides a good estimate of the change in the gap as between primes in the region of n.

So as we have seen as we move for example from 1,000,000 to 1,000,001 the change in the gap would be close to 1/1,000,000.

Now the extent to which the harmonic series falls short in terms of predicting the actual gap between primes is explained by Euler's Constant (= .5772...).

More accurately the gap between primes (in the region of n) is given by log n.

And as the harmonic series approximates log n + λ (where λ = Euler's Constant) then using the harmonic series as a prediction will overestimate the gap by .5772.


However remarkably the value of Euler's Constant is itself related to all other positive integer values of the Zeta function.

So λ = ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +.....

Therefore when we confine ourselves to finite values of n, the value of log n (representing the average gap between primes) can be expressed in terms of the positive integer values for s of the Zeta Function

i.e. ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -......


We can likewise modify the initial estimate of the change in average spread between primes (in the region of n).

So now using all of these Zeta values a better approximation is given by

1/n - 1/{2*(n^2)} + 1/{3*(n^3)} - 1/{4*(n^4)} +.....

We could also express the above as the value of log (n + 1) - log n


However as n becomes very large the modifications to the initial estimate of the change in spread (i.e. 1/n) are so small as to be negligible. Therefore 1/n serves as a particularly good estimate!


We can also express log n in another way as the sum of Zeta function values for positive integer values of s > 1.


Therefore the average gap between primes is given as

ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +.....


So n/log n representing the frequency of primes (among the first n natural numbers)

→ n/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}

or alternatively,

→ n/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}


As we have seen n in fact represents the value ζ(0) where the sum of terms is taken over a finite limited range.

Therefore

n/log n → ζ(0)/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}

or alternatively,

n/log n → ζ(0)/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}