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The Theory of the Riemann Zeta-Function

E. C. Titchmarsh, revised by D. R. Heath-Brown
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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Despite its age this is probably the most-cited book on the zeta function. The book has a long history, starting out in 1930 as a Cambridge Tract and issued in 1951 as a much-expanded monograph. The present volume, published in 1987, was created from the 1951 edition by adding a large number of chapter-end notes by Roger Heath-Brown. The notes cite the most recent results, but in general do not give proofs.

The book is very comprehensive, and covers nearly anything anyone would want to know about the zeta function. There are two fairly lengthy chapters on applications to the Prime Number Theorem and the Dirichlet divisor problem. The zeta function is still an active area of research, particularly the distribution of its zeroes and the Riemann hypothesis, although most of the real breakthroughs had already happened by 1986 and are at least mentioned in the notes. The last chapter, on calculations of the zeroes, is very brief, and is mostly conceptual; there is a very brief note about more recent computer work. Very Bad Feature: no index!

The present book and Ivić’s 1985 The Riemann Zeta Function: Theory and Applications are almost the same age and cover about the same topics, and both are good reference works. Ivić has the advantage of being written from scratch and it does contain proofs for the more recent results. Titchmarsh is slanted slightly toward growth estimates, and Ivić is slanted slightly toward moment theorems and estimates of the density of the zeroes. Neither book is very good for beginners, partly because they are so detailed and partly because they are (for the most part) far removed from the applications. Both are monographs and have no exercises. Two good short introductory books are Patterson’s An Introduction to the Theory of the Riemann Zeta-Function and Iwaniec’s very brief Lectures on the Riemann Zeta Function. Edwards’s Riemann’s Zeta Function is also a very good introduction and includes a good historical viewpoint, but is not especially concise.

For someone interested in getting started with the zeta function, I would not recommend any of these books. It would be better to start out with a number-theory book that presents a complex-variables proof of the Prime Number Theorem, because that shows clearly why the zeta function is important and further which of its properties are important. I think Ingham’s The Distribution of Prime Numbers is an especially good introduction from this viewpoint, because it weaves the number theory and zeta function aspects together very skillfully. True, it is very old, having been published in 1932, although the current reprint includes a 1990 Foreword by R. C. Vaughan bringing some of results up to date. The recent Number Theory: An Introduction via the Density of Primes by Fine & Rosenberger is completely up-to-date and gives a lot of context; it uses the very slick 1980 proof by D. J. Newman.

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.

I. The Zeta Function and the Dirichlet Series Related to It

II. The Analytic Character of Zetas and the Functional Equation

III. The Theorem of Hadamard and De La Vallée Poussin, and Its Consequences

IV. Approximate Formulae

V. The Order of Zetas in the Critical Strip

VI. Vinogradov’s Method

VII. Mean-Value Theorems

VIII. \(\Omega\) Theorems

IX. The General Distribution of Zeros

X. The Zeros on the Critical Line

XI. The General Distribution of Values of Zetas

XII. Divisor Problems

XIII. The Lindelöf Hypothesis

XIV. Consequences of the Riemann Hypothesis

XV. Calculations Relating to the Zeros