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The Riemann Zeta-Function: Theory and Applications

Aleksandar Ivić
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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As readers of Greg Chaitin's article know, it's zeta function time. Mathematicians who read one of the recent popular accounts of the Riemann Hypothesis might be interested in learning more. So Dover's decision to republish Aleksandar Ivić's The Riemann Zeta Function makes good business sense. It is also to be welcomed because this is a useful book.

There are not all that many books that focus exclusively on the Riemann zeta function. H. M. Edwards' Riemann's Zeta Function, also from Dover, comes to mind first. Edwards focuses on Riemann's original paper, using it as a springboard to develop the modern theory. I am aware of two other books: a more recent (1988) treatment by S. J. Patterson and the updated classic by E. C. Titchmarsh (revised by D. R. Heath-Brown). Ivić's book dates back to 1985, so it does not include the most recent results (e.g., there is nothing about random matrices), but it does offer a thorough treatment of the theory.

I'm not enough of an expert to offer advice on which of these books is the best reference on the zeta function. (The best entry point for the non-specialist is probably a broader book on analytic number theory, from which one might proceed to one of these books.) As a part-time historian, I tend to lean towards Edwards, but Ivić covers much more ground. In any case, I'm delighted to see this affordable Dover edition of a book that had been out of print.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME. He is interested in number theory, poetry, the history of mathematics, and football (the real thing, not the American version).


1. Elementary Theory
2. Exponential Integrals and Exponential Sums
3. The Voronoi Summation Formula
4. The Approximate Functional Equations
5. The Fourth Power Moment
6. The Zero-free Region
7. Mean Value Estimates Over Short Intervals
8. Higher Power Moments
9. Omega Results
10. Zeros on the Critical Line
11. Zero-density Estimates
12. The Distribution of Primes
13. The Dirichlet Divisor Problem
14. Various Other Divisor Problems
15. Atkinson's Formula for the Mean Square
  Appendix. Indexes