In
yesterday’s blog entry, I concluded by illustrating that the average spread
between primes (to n) is complemented by the average frequency of natural
number factors (per unit).

This in
fact points to the synchronous nature of the number system where complementary
aspects of behaviour dynamically interact with each other.

Put another
way, the behaviour with respect to the primes regarding the average spread
between each member at various intervals of the number system intimately
depends on the corresponding behaviour with respect to the (average) frequency
of natural number factors over these same intervals.

Equally, in
reverse, the behaviour with respect to frequency of the natural number factors
intimately depends on the corresponding behaviour with respect to the average
spread as between the primes.

In other
words both aspects of number behaviour co-determine each other in a holistic
synchronistic manner.

However to
properly appreciate the qualitative nature of such synchronistic behaviour, we
must inherently view the number system in a dynamic manner (representing again
the interaction of complementary opposite poles of behaviour).

Now the
harmonic series,

1 + 1/2 +
1/3 + 1/4 + …..1/n = log n + γ (where γ is
the Euler-Mascheroni constant = .5772… approx).

Once again,

log n – log
(n – 1) = 1/n + 1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……
Therefore
when n = 2,

log 2 – log
1 = 1/2 + (1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……^{ }) where again n = 2.
Likewise,

log 3 – log
2 = 1/3 + (1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……) where again n = 3
and

log 4 – log
3 = 1/4 + (1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……) where again n = 4
and
continuing on in this fashion, finally,

log n – log
(n – 1) = 1/n + (1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……) where again n = n.
Therefore
summing up terms on both LHS and RHS

log n – log
1 = 1/2 + 1/3 + 1/4 + …… 1/n + ∑(1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4 }+……) where the value of n is taken from 2 to n.
Therefore
because log 1 (in real terms) = 0, then

log n = log
n + γ – 1 + ∑(1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4 }+…… ).
Therefore γ
= 1 – ∑(1/2n

^{2 }+ 1/3n^{3 }+ 1/4n^{4}+……), again summed from 2 to n.
So this
provides one interesting way of expressing the value of the Euler-Mascheroni
constant!

Interestingly
when n = 1, the Riemann Zeta Function i.e. ζ(1) (which results in the harmonic
series) and the expression for log n – log (n – 1) are identical.

Indeed γ
can equally be given a fascinating expression in terms of the Riemann Zeta
Function so that,

λ →
ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +….. (when summed to a finite number of terms
with approximation improving as the number of terms increases).

This would imply therefore that

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

Alternatively,

λ → ζ(1) –
{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..} again when summed to a finite n, with
approximation improving as n improves.

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