Book Review: Two Books on Analytic Number Theory

Analytic number theory is the application of methods from analysis to the study of integers, in particular primes. This may seem paradoxical: at the heart of analysis lie notions of continuity and differentiability — and what could be more discrete and discontinuous than the set of primes?

The first step in connecting these two domains is the Euler Product: an identity that relates an (infinite) product over the primes to an analytic function:

$$ \prod_\text{all primes $p$} \left( 1 - \frac{1}{p^s} \right)^{-1} = \sum_\text{all positive integers $n$} \frac{1}{n^s} \qquad \text{where $s$ is any complex number} $$

The expression on the right is the definition of tge Riemann Zeta Function, and much of analytic number theory is devoted to the study of this function. It also the subject of the Riemann Hypothesis: the hypothesis that all of its “non-trivial” zeros have a real value of $1/2$.

This is non-trivial stuff, and it may seem incredible that much of it can be made accessible at an early-undergraduate level. But the following book does an outstanding job at it.

A Primer of Analytic Number Theory

  • Jeffrey Stopple
  • 400 pages
  • Cambridge University Press (2003)
  • ISBN: 978-0521012539

This book begins very simply, studying some ancient identities among integers. It then moves on to sums and series, focussing increasingly on prime numbers. Eventually, the Riemann Zeta function is introduced via Euler’s Product. The author then walks the reader through the various analytical operations that lead to Riemann’s Explicit Formula: an explicit formula (in terms of sums and integrals), which yields a function that counts the number of primes below its argument. The book concludes with outlooks on Diophantine Equations and Elliptic Curves.

The book develops most of the non-elementary methods it uses as it goes along, beginning with infinite series, then introducing modular arithmetic, and complex numbers. Having a background in these topics surely helps, but is not strictly required, making this book accessible even to readers with a rather modest, or passive, mathematical background.

Of course, even this treatment can’t avoid some of the technicalities that are characteristic of this field; in particular in the second half, the reader needs to work a little harder to follow the argument along. Nevertheless, all the details are worked out with great clarity and in sufficient detail.

If one is looking for a mathematical and rigorous (as opposed to popular), yet entirely accessible introduction to Analytic Number Theory, this book is a terrific resource.

Making Transcendence Transparent

  • Edward B. Burger, Robert Tubbs
  • 272 pages
  • Springer Verlag (2004)
  • ISBN: 978-0387214443

This book is an introduction to the theory or Transcendental Numbers, that is, numbers that are not the solution of an algebraic equation. This is a fairly difficult and technical subject, and this book requires significantly deeper background, and more effort, then Stopple’s book.

Nevertheless, if one is interested in developing at least a sense for the type of arguments that are used in classical Transcendental Number Theory to establish the transcendence of $\mathrm{e}$, $\pi$, and $\mathrm{e}^{\pi}$, then this is a good starting point. The authors point out early that the idea of these proofs is basically always the same. Transcendence proofs are usually indirect: to prove that a number $x$ is transcendental, assume that it is rational. Then construct a polynomial with integer coefficients, such that $x$ is a root of that polynomial. Then construct an integer from the coefficients in this polynomial and use analytical methods to show that this integer is simultaneously greater than zero and smaller than one, hence creating a contradiction.

The authors style is entertaining, and even humorous, and they manage to walk the reader through their often long and technical proofs with grace and style.