# Russell Sears Reviews Stalking the Riemann Hypothesis

November 2, 2006 |

Dan Rockmore’s book Stalking the Riemann Hypothesis, written for the recreational mathematician, brought forth in me the profound emotions. Reading it seemed quite similar to my enlightenment in reading my first Calculus book. The book completely avoided the messy math and stuck to the core concepts behind the math. And what a mathematical journey is was. Starting with the early history of pure math, with the Greeks, to modern day theories. It seems that search to prove Riemann’s hypothesis has lead to many the leading edge of science. Chaos theory, Fractals, the edge of probability theory, Random infinite matrices and more.

Besides the beautiful ideas it goes introduces you to many of the great minds that have worked on this problem, and are working on this problems. Two quotes that just floored me:

Eugene Wigner: “The unreasonable effectiveness of mathematics” (sounds like a counter to me)

And for those convinced they can’t ever get through the math. Polya in his book “How to Solve It”

“If you cannot solve a problem then there is an easier problem that you can solve. Find it.”

These alone make it worth the read. You find all sorts, much like you would in our field. The magician turned mathematician. The hungry kid getting a break. Each story is inspiring. Some you probably are familiar with but, Dan writes very interesting facts on each.

But beyond this, it has a profound impact due to the underlying philosophical impact. The hope that we are beginning to understand the meaning to randomness, and through this perhaps a light showing hidden secrets of our universe, in the infinite and infinitesimal.

For those of you not familiar with the Riemann Hypothesis, it is one of the most studied areas in pure math, due to its relevance to the prime numbers. Before Riemann expanded it through analytic continuation to the complex plane, Euler expressed it as for any S > 1 as = 1/(1^s)+ 1/(2^s) + 1/(3^s) + 1/(4^s)+ 1/(5^s)…. The relationship to the primes can be seen by its equivalence to Product of 1/(1-p^-s). Though not in the book, this proof is quite beautiful and can be seen sieving out the primes.  The hypothesis is that the Reimann zeta function, the expansion to the complex plane, only has non-trivial zeros on the real line 1/2. (The trivial zeros are multiples of -2). With a little imagination and idea of what the Fourier transform does, the Fourier waves turn at the prime to give a estimate for any number how many prime their are before it. The math is beautiful, the writing is great, and the spirit is uplifting.

Vincent Andres responds:

Concerning didactic books and Fourier waves, a very didactic and commendable book is The World According to Wavelets, by Barbara Burke Hubbard.  The title of the book is about wavelets, but in fact half of the book is about Fourier, in a very didactic way.