Tuesday, October 26, 2010

The Harmonic Series

The Pythgoreans have a crucial input into true appreciation of the Riemann Hypothesis in at least two important ways.

We have already discussed the first of these relating to the square root of 2 in "The Pythagorean Dilemma".

The Pythagoreans realised to their dismay that this number was irrational. It did not suffice for them to simply prove in quantitative terms that the number was indeed irrational. More importantly they were seeking a qualitative appreciation as to how such a number could arise.

The essence of an irrational number is that it contains both finite and infinite aspects. In the subsequent mathematical understanding however of irrational numbers a solely reduced quantitative interpretation is provided. Thus the real need, which the Pythagoreans clearly realised, to provide both quantitative an qualitative interpretation has thereby been avoided.

The square root of 2 also points to a circular appreciation of number where - paradoxically - both positive and negative interpretations can both be correct.
So correct to 4 decimal places the square root of 2 can be expressed as + 1.4142 or - 1.4142.

Now the relevance of all this to the Riemann Hypothesis is that we obtain the square root by raising 2 to the dimensional power of 1/2.

And of course the Riemann Hypothesis states that all the non-trivial zeros of the zeta function will relate to values of s i.e dimensional powers with real part = 1/2.

However the Pythagoreans became equally famous for the discovery of the significance of what has come to be known (in their memory) as the harmonic series.

This is the simple sequence comprising the reciprocals of the natural nos.

1 + 1/2 + 1/3 + 1/4 +.....

Now the Pythagoreans were able to connect this series with musical harmonics. They seemed to realise that if a vessel full of water was struck and then successively also struck when half full, a third full, a quarter full etc. that the musical notes generated would appear harmonious to the ear.
Indeed this provided for them striking confirmation of the overriding importance of the natural numbers in explaining nature's secrets. So the very terms "music of the spheres" derives from this discovery.

However the harmonic series provides the starting base for - what is known as - Riemann's Zeta Function where each of the natural number reciprocals can be raised not alone to the dimensional power of 1 but to any complex number power.

However the harmonic series in itself can be shown to have a striking relevance for understanding the behaviour of the primes.

The prime number theorem is often stated in its simplest form as where the ratio of n/log n to the true frequency of primes approaches 1 as n becomes progressively larger.

This implies that log n provides a very good measurement (especially for large values of n) of the average gap as between successive prime numbers.

However there is a close connection as between the harmonic series and log n.

In fact the harmonic series (where the denominator ranges over the natural numbers from 1 to n) = log n + k (where k is known as Euler's constant = .5772 approx).

Thus for very large values of n the sum of the harmonic series itself provides a very accurate estimate of the average gap between successive prime numbers.

So this provides just one striking example as to the close connection as between the primes and natural numbers!

It is also remarkable in another sense. Once again the sum of the harmonic series (where n is finite) = log n + k.

Thus when we differentiate with respect to n we get 1/n.

This therefore implies that the average gap as between primes increases by 1/n (as n increases by 1)

This would imply for example that as move from n = 1,000,000 to 1,000,001 that the average gap between primes itself increases by 1/1,000,000 (or perhaps more accurately 1/1,000,000.5).

This is truly remarkable in that it links an important aspect of prime number behaviour in an extremely simple manner to the reciprocals of the natural numbers!

It is also fascinating that the famous formula n/log n relating to the frequency of the primes can be expressed as the ratio of the two zeta functions (where n is taken over a finite range) for s = o and s = 1 respectively.

When s = o, the zeta function

= 1/1^0 + 1/2^0 + 1/3^0 + 1/4^0 + .....

= 1 + 1 + 1 + 1 +...... = n

When s = 1, the zeta function

= 1/1^1 + 1/2^1 + 1/3^1 + 1/4^1 + ..... (which approximates to log n when n is suitably large)

Therefore we can approximate n/log n as

zeta (0)/zeta (1) where the range of values for n is finite (and values of series calculated in the conventional linear manner).

So perhaps (though more inaccurate for lower values of n) this could provide the simplest formulation of the prime number theorem,

i.e. {zeta (0)/zeta (1)}/{actual occurrence of primes from 1 to n} approximates 1(when n is sufficiently large in magnitude).

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