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The Distribution of Prime Numbers

A. E. Ingham
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Mathematical Library
[Reviewed by
Allen Stenger
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This is a concise introduction to the ideas behind the prime number theorem (PNT), as they stood in 1932. True, that was a long time ago, but really the basic ideas haven’t changed that much since then. If this book were done over today, it would probably have more about sieves and maybe would include an elementary proof of the PNT. It likely would have more about primes in arithmetic progressions, a subject that is barely touched on in the present book. In fact Tenenbaum & Mendès-France’s The Prime Numbers and their Distribution might be considered a modern version of the present book, although I find it harder to follow (especially the elementary proof of the PNT).

The original publication was a Cambridge Tract, and it follows their usual focused pattern of condensing a great deal of important information into a short but clear package (just over 100 pages in this case). The current reprint includes a 1990 Foreword by R. C. Vaughan bringing some of the results up to date.

The book covers most of the important elementary results (infinitude of primes, Chebyshev estimates, Mertens’s theorems). It then gives a very clear, although slightly antiquated, proof of the PNT. The antiquated part is that the proof still depends on some growth estimates for the Riemann zeta function. The Wiener-Ikehara proof, which came out just as the book was being written, was the first to base the proof solely on the non-vanishing of zeta function on the line with real part 1. There have been several slicker proofs developed since then (D. J. Newman’s 1980 proof, as polished by Don Zagier, is my favorite). The proof in the present book is especially nice because it shows clearly how properties of the zeta function translate into properties of the primes; it also has an especially simple form of the Tauberian argument, and so reveals why this kind of argument is important.

The book then goes on to develop a larger zero-free region of the zeta function, and shows how this is used to get an improved error term in the PNT. This is an area that has been worked ever since the PNT was first proved in 1896, so there are better results available today. The exposition here, while not up-to-date, does give the reader a good understanding of how these arguments work. The same is true for the next chapter, on explicit formulae. The last chapter covers irregularity of distribution, that is, lower bounds for the error terms. This is one area that has changed a lot since the book was published, and where the book is probably less useful today.

I especially like this book because it weaves the number theory and zeta function aspects together very skillfully. It’s a very good introduction for someone who wants to learn about the zeta function or about prime number theory.

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.



1. Elementary theorems

2. The prime number theorem

3. Further theory of \(\zeta(s)\). Applications

4. Explicit formulae

5. Irregularities of distribution