Interesting Log Relationships (2)

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 + 1/2n² + 1/3n³ + 1/4n⁴ +……

Therefore when n = 2,

log 2 – log 1 = 1/2 + (1/2n² + 1/3n³ + 1/4n⁴+…… ) where again n = 2.

Likewise,

log 3 – log 2 = 1/3 + (1/2n² + 1/3n³ + 1/4n⁴ +……) where again  n = 3

and

log 4 – log 3 = 1/4 + (1/2n² + 1/3n³ + 1/4n⁴ +……) where again n = 4

and continuing on in this fashion, finally,

log n – log (n – 1) = 1/n + (1/2n² + 1/3n³ + 1/4n⁴ +……) 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 + ∑(1/2n² + 1/3n³ + 1/4n⁴ +……) 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 + ∑(1/2n² + 1/3n³ + 1/4n⁴ +…… ).

Therefore γ = 1 – ∑(1/2n² + 1/3n³ + 1/4n⁴ +……), 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.