Someone unfamiliar with number theory might find the title of Paul Pollack’s excellent book on the subject puzzling: Not Always Buried Deep: A Second Course in Elementary Number Theory. The first part of the title is easy enough to explain. Many theorems in number theory are indeed “buried deep.” A result may be very difficult to prove even though it is easy to state. However, as Pollack points out, this is not always the case. Some profound results can be established relatively easily.
The second part of the title requires more explanation. How can the second course in anything be elementary? In common parlance, you’d expect the second course in a subject to be “advanced,” or at least “intermediate.” However, number theory has a technical idea of what is “elementary.” Roughly speaking, “elementary” means “not using advanced mathematics, especially complex analysis.” Note that in this definition, the word “complex” is also used in a technical sense. “Complex analysis” refers to analytic function theory, the study of functions that are differentiable when the definition of derivative is extended to the complex plane. Elementary number theory can use analysis that is complex, but it cannot use complex analysis! Perhaps elementary number theory could be called “beeline” number theory: once you finish calculus, you make a beeline to number theory, learning only the general mathematics necessary along the shortest path to your goal.
The prime number theorem, the theorem that gives the asymptotic density of the primes, illustrates the difference between elementary number theory and analytic number theory. Not Always Buried Deep devotes about 20 pages to an “elementary” proof of the prime number theorem. The mathematics is subtle, but it doesn’t rely on complex analysis, and is elementary in a sense. In principle, a bright student could understand the proof armed with only a solid understanding of calculus. Hadamard and de la Vallée-Poussin independently proved the prime number theorem in 1896, but an elementary proof of the theorem didn’t come until over 50 years later in the papers of Selberg (1948) and Erdös (1949). In some ways the elementary proof was a greater achievement than the original analytic proof. Pollack says
… many prominent mathematicians, including Hardy, [believed] that such an elementary proof did not exist. When such a proof surfaced in 1948, it sent shockwaves through the world of mathematics.
Not Always Buried Deep presents those results from number theory that are not “buried deep,” i.e. that can be reached relatively quickly using elementary tools as defined above. The book also states a few deeper results without proof when necessary to complete a picture. It provides a gentle introduction to more advanced number theory. The mathematics is intricate at times, but not unnecessarily so. One gets the sense that Pollack took great care to make the material as accessible as possible.
Not Always Buried Deep would be a good text for a second undergraduate course in number theory. The book’s expository prose makes independent reading assignments feasible. It also provides numerous exercises at the end of each chapter. A course taught out of this book would provide a smooth transition to advanced study of number theory.
Pollack’s book is pleasant to read. The book appears to have been typeset in LaTeX just like countless other math books, but something about it makes the print more attractive. Perhaps it is the paper color; the pages have a subtle parchment tone. The book is also pleasant to read for its elegant mathematical content.
John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.