Analytic Number Theory

Donald J. Newman was a noted problem-solver who believed that math should be fun and that beautiful theorems should have beautiful proofs. This short book collects brief, self-contained proofs of several well-known theorems in analytic number theory, including Newman’s short analytic proof of the Prime Number Theorem. There are treatments of a number of other theorems whose proofs used to be formidable, including Waring’s problem, the non-vanishing of the Dirichlet L-series on the line Re s = 1, and an asymptotic formula for the number of partitions.

There are a few exercises at the end of each chapter, but these seem to be tacked on; their difficulty varies from trivial to almost impossible. The first printing of this book was loaded with typographical errors, to the point that it was almost impossible to follow at times. The second printing corrected nearly all of these, but introduced a few new ones; I have 24 typos marked in my copy of the second printing, in a 78-page book.

Most books today try to be comprehensive, even if much of what they comprehend is dull and uninspiring. There are surprisingly few books that deal with “fun proofs”. Perhaps the most notable recent example is Aigner and Ziegler’s Proofs from the Book, that covers all areas of mathematics and tries to present the most elegant proofs. In the area of number theory, the classical work of Landau, Elementary Number Theory, has much the same flavor, as it presents brief, self-contained solutions of several problems from classical number theory. A very recent book based on the same idea is Pollack’s 2009 Not Always Buried Deep: A Second Course in Number Theory, that like Newman’s book tries to show that many of the most famous results in number theory can be proved briefly and and with few prerequisites.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.