Saturday, January 21, 2012

Music of the Primes

As we know Marcus du Sautoy gave the title "The Music of the Primes" to his book on the Riemann Hypothesis.

Interestingly, I see that Keith Devlin gives the same title to his Chapter on "The Riemann Hypothesis" in his book "The Millennium Problems".

It started with the Pythagoreans that discovered that the terms in the series 1,. 1/2, 1/3, 1/4, .. have a striking relevance for musical sound.
So this series was called the harmonic series.

This very series then provided the base for what ultimately was to become the Riemann Zeta Function which was shown to have a special relevance for the distribution of the primes. And then Riemann showed how to magically correct the general distribution of the primes with the wavelengths associated with the famed non-trivial zeros of the Zeta Function so as ultimately predict the exact number of primes up to a given number.

So one again there is a close relationship as between these wavelengths and musical harmonics.

So it is not perhaps surprising therefore that the term "the music of the primes" has proven so popular.

However the point I would make is that whereas prime numbers are conventionally interpreted in a quantitative manner, clearly the musical connotation with respect to the overall behavior of the primes has a qualitative (rather than quantitative) significance.

So therefore, properly understood the primes have both quantitative and qualitative aspects (which are interdependent).

Thus the key issue therefore with respect to the primes relates to the ultimate reconciliation of the quantitative with the qualitative aspect.

Once more as Conventional (Type 1) Mathematics relates merely to the quantitative aspect, it cannot satisfactorily hope to explore the true mystery of the primes.

Indeed as so often stated on this blog, the Riemann Hypothesis relates to the key condition necessary for the ultimate identification of both the quantitative and qualitative aspects of the primes. As the qualitative aspect is formally ignored by the mathematical profession, the Riemann Hypothesis cannot be proved (or disproved) in conventional terms. Indeed it points to an ineffable physical state which can only be fully understood in a corresponding ineffable spiritual manner.

And in this state the true mystery of the primes ultimately resides!

1 comment:

  1. Riemann hypothesis is about the physics of arithmetic . I think I have devised a way to solve it using physical invariancy of graph (given the zeta function is complex-valued and the evolution of complex-valued function shows the physical invariancy under transformational mapping across the mathematical domains). Using this invariancy law from physics governing complex-valued functions,I have shown that locus of non-trivial zeros and trivial-zeros must be collinear and thus lie on the geometrical bisectors of the physical spaces of mathematical domains along their respective degrees of freedom in Euclidean/non-Euclidean metric space of the mathematical domains. By the way your QUALITATIVE version is also one of the great methods but unfortunately mathematical community doesn't recognize yet.

    Pankaj Mani,India (!/pankaj.mani?sk=wall)