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PI and the AGM: A Study in Analytic Number Theory and Computational Complexity

Jonathan M. Borwein and Peter B. Borwein
John Wiley
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a very erudite book, and a large part of its charm is that it shows us many unexpected connections between different parts of mathematics. The AGM of title is the Arithmetic-Geometric Mean, a recurrence that was initially studied by Lagrange and Gauss in connection with numerical approximations to elliptic integrals. The book is inspired to some extent by Ramanujan’s 1914 paper, “Modular Equations and Approximations to Pi”. Like much of Ramanujan’s work, this paper is full of interesting ideas but skimpy on proofs, and the present book is the first time that these ideas were worked out in detail and proved.

The publisher classifies this book as number theory, and there is some truth to that. It also deals with many of the topics of analysis from the late 1800s and the early 1900s, in particular theta functions and modular groups, and the last half of the book is concerned mostly with numerical approximations for various transcendental functions, and calculations for \(\pi\) are spread throughout the book.

This summary makes the book sound like a hodgepodge, but in fact it hangs together very well. Roughly the first half of the book deals with elliptic integrals, theta functions, and the AGM. This part includes materials on partitions, the Rogers–Ramanujan identities, and representing numbers as sums of squares. It also includes the formulas and algorithms for calculating \(\pi\). The second half of the book is focused more on computer arithmetic and complexity of calculation, including results on fast Fourier transforms and fast multiplication. It also includes the history of \(\pi\) calculations and results on diophantine approximations (irrationality measures) of \(\pi\). About 1/4 to 1/3 of the book is exercises, so it would work well as a textbook or problem book as well as a monograph (though there are no solutions; some exercises have hints).

Jonathan Borwein (1951–2016) was fascinated by \(\pi\), and there is much additional information on all aspects of \(\pi\) in the two source books he co-edited: Pi: A Source Book and Pi: The Next Generation. An excellent introduction to theta functions is Bellman’s A Brief Introduction to Theta Functions. This is another idiosyncratic book that has some of the same flavor as the present book, although it’s not as diverse.

Bottom line: the book is a tour de force. If you are interested in any of topics covered, it will lead you along somewhat-winding paths to other topics you’ll like. Even if you’re not interested, take a look and marvel at how closely connected its many diverse topics are.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration.

Theta Functions and the Arithmetic-Geometric Mean Iteration.

Jacobi's Triple-Product and Some Number Theoretic Applications.

Higher Order Transformations.

Modular Equations and Algebraic Approximations to pi.

The Complexity of Algebraic Functions.

Algorithms for the Elementary Functions.

General Means and Iterations.

Some Additional Applications.

Other Approaches to the Elementary Functions.



Symbol List.