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Number Theory, Fourier Analysis and Geometric Discrepancy

Giancarlo Travaglini
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 81
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This book is a self-contained look at several problems and techniques in the geometry of numbers. The Fourier analysis slant comes from the fact that the book uses Fourier series and integrals, the Poisson summation formula, and some general exponential sums to approximate the discontinuous functions of number theory and produce some asymptotic formulas. Discrepancy theory originates in the uniform distribution of sequences, as a way of quantifying the worst cases: Even if a sequence is uniformly distributed mod 1, this is only true in the limit, and finite portions of the sequence may deviate from the ideal distribution quite a bit. The idea is generalized to multiple dimensions and various shapes of approximating objects to get the theory of geometric discrepancy.

The book develops all the needed number theory and Fourier analysis, although it does assume a good command of multidimensional integration and some familiarity with Hilbert spaces. The book has a collection of reasonably-difficult exercises at the end of each chapter, so it is positioned as a stand-alone course at the upper-division undergraduate or beginning graduate level.

The model problems that run through the book are the Dirichlet divisor problem (the average number of divisors of an integer) and the Gauss circle problem (the average number of representation of an integer as the sum of two squares of integers). It’s not too hard to get the leading terms for these averages, so the challenge is to refine the error terms, and the book looks at several ways to do this. These are still active research topics, and the book gives representative treatments while quoting the latest results.

Bottom line: a well-written and easy-to-follow look at a variety of interesting topics in number theory, but with important tie-ins to analysis.

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

Part I. Elementary Number Theory:
1. Prelude
2. Arithmetic functions and integer points
3. Congruences
4. Quadratic reciprocity and Fourier series
5. Sums of squares

Part II. Fourier Analysis and Geometric Discrepancy:
6. Uniform distribution and completeness of the trigonometric system
7. Discrepancy and trigonometric approximation
8. Integer points and Poisson summation formula
9. Integer points and exponential sums
10. Geometric discrepancy and decay of Fourier transforms
11. Discrepancy in high dimension and Bessel functions