Sunday, April 02, 2017

Book Review: Prime Numbers and the Riemann Hypothesis

Prime Numbers and the Riemann Hypothesis by Barry Mazur and William Stein is a slender (142 pg.) book aimed at a varied audience of the mathematically curious. It is profusely illustrated, mainly with pictures of what the authors call the staircase of primes, a function that starts at zero and goes up by one each time a prime is encountered, though several recarpentried versions of the staircase also make the scene.

The book is divided into 38 very short chapters, organized into four sections, with the first and longest section (chapters 1-24) aimed at readers without a calculus background. The second section demands a bit of calculus (not much!) and the third some Fourier analysis, while the fourth gets to the nitty-gritty of the zeta function.

The figures and many of the calculations were done with Sage, a free mathware package developed by the second author, and made available to the eager experimenter.

The first section has a lot of the lore primes that is accessible at the elementary level, and that is a great deal. How many consecutive primes, for example, are separated by two (3-5, 5-7, 41-43,...)? Nobody knows. How many are separated by an even number less than or equal to 246? That turns out to be known to be infinitely many.

This isn't a textbook, and doesn't have problems, as such, but there are a few "you might try proving" suggestions. Here is the first one, a fairly good test of your basic algebra (or at least mine): A number of primes have the form 2^p - 1. Show that if p is not prime, then 2^p - 1 is composite (not prime). If that's too tough, try this: How many pairs of consecutive primes are separated by an odd number? ;-)

Along the way, we meet several different incarnations of the Riemann Hypothesis, the first one being: For any real number X the number of prime numbers less than X is approximately Li(X) and this approximation is essentially square root accurate. Here Li(X) is the log integral of X = Integral[(1/Log(t))dt, {t,0,X}]. (by Log we mean natural Log)

Sections II and III of the book are devoted building up the apparatus needed to transform this statement into Riemann's form, which looks superficially very different: All the non-trivial zeroes of the zeta function lie on the vertical line in the complex plain consisting of the complex numbers with real part 1/2. These zeroes are (1/2 plus or minus i*theta(i)) where the theta(i) comprise the spectrum of primes talked about in the earlier chapters.

Despite a good deal of verbiage devoted to the subject in the earlier chapters, I was never quite clear on exactly how these values are calculated, though I think that they are the Fourier transform of some version of the staircase of primes. I'd just like an equation that said theta(i) = some expression.

The first author and I share same last name (though spelled differently). Though that name is quite rare, I have no reason to believe that we are related.