An Introduction to the Theory of the Riemann Zeta-Function

This is a clear and concise introduction to the zeta function that concentrates on the function-theoretical aspects rather than number theory. It does include one chapter on the Prime Number Theorem, focusing on error estimates and on the relation between the error estimates and the location of the zeroes of the zeta function, rather than on the simple asymptotic form of the theorem. The author describes the book’s purpose (p. xi) as, “…to bring out the central role of the Poisson Summation Formula, and, especially, of the ‘explicit formulae of prime number theory’.”

I especially like the author’s assessment of the prospects for the Riemann Hypothesis (beginning on p. 74). He starts by saying, “We now turn to the question as to why one might expect the Riemann Hypothesis to hold. The reasons are not as compelling as one might like to think.” Then he lays out a fair case both for and against. I have to admit after having studied it that the evidence in favor is flimsy.

The exercises are especially good, numerous and challenging. They extend the results of the text, or ask you to prove analogous results.

Very Good Feature: Seven appendices that give most of the function-theoretical background you need to know to read this book. The Fourier Theory appendix is a gem: everything you need to know about the subject, including proofs, in 11 pages!

The standard works in this field are Titchmarsh’s The Theory of the Riemann Zeta Function and Ivić’s The Riemann Zeta Function: Theory and Applications. The present work differs from those mostly in being shorter and not going into as much depth. I think it is also easier to read than those, both because the writing is better and because of the helpful appendices. Advances continue to be made in our knowledge of the zeta function, and none of these books (all 20 to 30 years old) is very up-to-date today, but the present book is still very valuable as an introduction.

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 allenstenger.com. His mathematical interests are number theory and classical analysis.