The Harmonic Series Again

Euler made great advances with respect to better understanding of the prime numbers.

In what must constitute one of the most truly memorable contributions to Mathematics he was able to demonstrate an intimate connection as between the natural numbers on the one hand and the prime numbers on the other.

Now once again the zeta function is defined as ∑1/n^s (for n = 1 → ∞)

The harmonic series results from setting s = 1

Thus ∑1/n = 1 + 1/2 + 1/3 + 1/4 +...... which is divergent.


Now the Euler zeta function is defined for values of s (> 1) with

∑[1/n^s] = ∏{p^s/(p^s – 1)} where again p ranging from 2 → ∞


Therefore when s = 2

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +... = {2^2/[(2^2) - 1]}*{3^2/[(3^2) - 1]}*{[5^2/([(5^2) - 1]}

So

1/1 + 1/4 + 1/9 + 1/16 +..... = 4/3 * 9/8 * 25/24 *.....


So we have here an intimate connection as between the natural numbers on the one hand (connected through addition)and the prime numbers (connected through multiplication).

Furthermore a unique dimensional connection exists between the two for all numbers where s > 1.


The question then arises as to whether a unique connection also exists in the vitally important case where s = 1.

Clearly in this case both the natural number sum series and the prime number product series will both diverge.

However if we confine ourselves to a limited finite number of terms, then an interesting connection does indeed exist.

So where the harmonic series is summed to a finite number of terms (n)

∑1/n → ∏{p + 1)/p}


So

1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → 3/2 * 4/3 * 6/5 *.......

So as we obtain the sum of the first n terms of the harmonic series on one side, we approximate the corresponding product of all p terms up to n on the other side.


This relationship can also be written in another interesting way!

1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → (1 + 1/2) * (1 + 1/3) * (1 + 1/5) *.....

Also when the no. of terms on the LHS = n, the corresponding no. of terms on the RHS approximates n/log n.


Thus on the LHS we have the sum of terms entailing the reciprocals of the natural nos (starting with 1); on the RHS we then have the product of terms entailing the reciprocals of the primes (in each case added to 1).


Now what this formulation clearly demonstrates is the intimate relation as between addition and multiplication in the connection of the primes to the natural number system.

However though - correctly - understood, multiplication involves both a quantitative and qualitative transformation with respect to number interpretation, Conventional Mathematics is based on the merely quantitative aspect. Therefore it inevitably reduces the qualitative aspect to the quantitative.

This quite simply then constitutes not alone the key barrier to solving the Riemann Hypothesis but in fact likewise the key barrier to its proper interpretation.

For ultimately the Riemann Hypothesis relates to the reconciliation of both the quantitative and qualitative aspects of Mathematics.

So again putting it bluntly, not alone can the Riemann Hypothesis not be solved in a conventional mathematical fashion, it is not even capable of being properly understood in this manner!