Thursday, June 21, 2012

Prime Number Theorem: Alternative Formulation

In a short addendum (on the "The Euler Identity" blog), a simple approximation for log n (with n positive)was provided i.e.

log n → (n^x - 1)/x as x → 0.

The Prime Number Theorem (providing the general distribution of the primes among the natural numbers) in turn can be simply expressed by the expression n/log n (as n → ∞).

Therefore substituting our approximation for log n, the Prime Number Theorem could be expressed as nx/(n^x - 1) as n → ∞ and x → 0!

It is often expressed that the secret of prime number behaviour lies in the relationship as between addition and multiplication!

It could equally be said that this secret lies in the relationship as between multiplication and exponentiation (which is simply demonstrated by this expression).

Also we have the combination of two extremes whereby the relative approximation is continually improved through making one variable (n) progressively larger, while the other variable (x) is made progressively smaller.

This likewise captures the essence of prime number behaviour which represents an extreme as between linear (quantitative) and circular (qualitative) aspects, whereby the relative independence of each prime number (as discrete) is made compatible with the overall interdependence of prime numbers (as continuous).

We also had provided a complementary Prime Number Theorem where the average absolute value of prime numbered roots of 1 approximates to 2/π = i/log i.

Once again i/log i can be approximated as ix/(i^x - 1) as x → 0.

So 2/π → ix/(i^x - 1) as x → 0.

1 comment:

  1. Interesting discussion. There is indeed a spiritual dimension to prime numbers. I've been literally moved to tears grocking the beautiful symmetries revealed when studying factorization sequences at the digital root level shaped by the so-called 'Trinity of Triangles' derived from the first 24 elements in the set of natural numbers not divisible by 2, 3 and 5. Not only are the primes not random, the deterministic algorithm that ultimately determines their sequencing is a beautifully symmetrical mathematical object. As evidence, consider the Magic Mirror Matrix: