Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

March 21, 2016

Prime Numbers and the Riemann Hypothesis

Posted by John Baez

I hope this great book stays open-access, but I urge everyone to download a free copy now:

It’s the best elementary introduction to the connection between prime numbers and zeros of the Riemann zeta function. Fun, fun, fun!

The preface gives an idea of what this book is like:

The Riemann Hypothesis is one of the great unsolved problems of mathematics, and the reward of $1,000,000 of Clay Mathematics Institute prize money awaits the person who solves it. But — with or without money — its resolution is crucial for our understanding of the nature of numbers.

There are several full-length books recently published, written for a general audience, that have the Riemann Hypothesis as their main topic. A reader of these books will get a fairly rich picture of the personalities engaged in the pursuit, and of related mathematical and historical issues.

This is not the mission of the book that you now hold in your hands. We aim – instead — to explain, in as direct a manner as possible and with the least mathematical background required, what this problem is all about and why it is so important. For even before anyone proves this hypothesis to be true (or false!), just getting familiar with it, and with some of the ideas behind it, is exciting. Moreover, this hypothesis is of crucial importance in a wide range of mathematical fields; for example, it is a confidence-booster for computational mathematics: even if the Riemann Hypothesis is never proved, assuming its truth (and that of closely related hypotheses) gives us an excellent sense of how long certain computer programs will take to run, which, in some cases, gives us the assurance we need to initiate a computation that might take weeks or even months to complete.

To inspire the reader — and I hope that means you! — Mazur and Stein give two interesting quotes. The first is by Peter Sarnak:

The Riemann hypothesis is the central problem and it implies many, many things. One thing that makes it rather unusual in mathematics today is that there must be over five hundred papers — somebody should go and count — which start ‘Assume the Riemann hypothesis,’ and the conclusion is fantastic. And those [conclusions] would then become theorems … With this one solution you would have proven five hundred theorems or more at once.

The second is by Don Zagier. I find this more moving, because of trying to wow us with hundreds of undescribed theorems, it really gets to the heart of the matter.

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, [they are] the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.

Posted at March 21, 2016 5:12 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2872

1 Comment & 0 Trackbacks

Re: Prime Numbers and the Riemann Hypothesis

I should add that if the first part of the book seems too elementary (it starts from the very basics), go to the end and work backwards.

Posted by: John Baez on March 22, 2016 4:56 PM | Permalink | Reply to this

Post a New Comment