Customer Reviews

The Prime Number Theorem (London Mathematical Society Student Texts)

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As an armchair mathematician, I can't remember how many times I'd read books that said things like "The prime number theorem, the proof of which is beyond the scope of this book, shows that the average number of primes less than any integer, n, is approximately n / log n." What a remarkable, intriguing statement! I was thrilled, therefore, to finally come across Professor Jameson's book, which is a proof of this theorem - no more, no less. Well, slightly more: he includes some interesting applications of the theorem, too.

The book is extremely well organized. It presents all necessary background material for the proof, and it does so in a refreshingly lucid manner. Topics are all well-motivated, and Jameson moves smoothly between them. He provides enough expository comments to guide the reader through the proof, but at heart this is a book of mathematics. I appreciated its utterly thorough proofs of all its statements, but those put off by equations will not enjoy this book.

Personally, the going got a little tough towards the culmination of the proof, around the midpoint of chapter 3. Those with a stronger background in analysis will doubtless find these sections easier to absorb.

Overall, this is a beautiful book. It clearly presents the theorem and the deep, subtle links between number theory and analysis. I highly recommend it.

This book is volume 53 of the London Mathematical Society Student Texts.
Both its inclusion in this series and its structure marks it as a textbook.
However, it subject matter is not what one would normally expect to find
as the sole subject matter of a textbook so I will break this review into
two parts. First, I will address what sort of textbook this is and where
it would sit in the University mathematics curriculum. Secondly, I will
review the contents of the book itself.

The author, G.J.O. Jameson was a mathematician at the University
of Lancaster in the UK. Thus it is natural to assume that he had the
British, rather than American, University system in mind when he wrote it.
The New Zealand University is based on the British system so I will outline
where it would find its place here.

The way the degree structure works is that all students
complete a three year undergraduate degree. The good students then
have a chance to enter the ``honours'' programme. By good we require
students to have completed 50 percent more courses than the minimum required for
the ordinary B.Sc. degree and to have obtained a minimum of a B+ average
in those courses. The B.Sc. Honours degree is a one
year post-graduate degree consisting of a number of courses and
a small research project. The emphasis changes between the undergraduate
and honours degrees. In the undergraduate degree each course aims to
provide the student with a broad level of appreciation, knowledge,
skill and understanding in the topic studied. At the honours
level the courses are focused on a narrow topic and are intended to
cover that topic in some depth. Thus an honours level course could well
be solely devoted to the proof and some applications of the Prime Number
Theorem. This textbook would provide an excellent resource for such a
course.

The book has a preface, six chapters, appendices A through H, a bibliography
and an index. The preface lays out in considerably more detail than I have
given above the author's motivation for writing this text, where it sits
in the University mathematics curriculum and how to use it in a course.
The appendices A through E cover some of the essential background mathematics
which will be required for successful study of the book. Anyone engaging
in self-study would be well advised to start with these appendices to
ensure you have the necessary mathematical background. The other three
appendices cover computation of the number of primes less than a given
number (appendix F), some tables of primes (appendix G) and some brief
biographical notes on some of the key mathematicians who contributed to
the proof of the Prime Number Theorem (appendix H).

The first two chapters take up almost 100 pages and are devoted to
developing the mathematics necessary to prove the Prime Number Theorem.
This is quite unlike any other textbook I have ever encountered. Usually a
text starts with some preliminaries and then sets about developing the
subject in a systematic way. Here the goal is the proof of the Prime Number
Theorem which requires a number of mathematical tools from different branches
of mathematics. Thus the chapters look like a rather eclectic collection
of small topics yet there is a organising principle here in that these are
needed to prepare us for the proof itself. A few examples; Abel summation,
Euler's summation formula, Dirichlet series, convolutions, the Mobius
function, the zeta function and series for log of zeta(s) among others.
In itself all of these topics are very interesting and should stimulate
a student's interest in these subject areas.

Initially it was not clear to me how the prime number theorem would be
proved but as I progressed through these two chapters I began to get
glimpses of how it might be done. So I believe these chapters prepare
the reader well for understanding the proof.

Chapter three gets down to the serious business of the proof itself.
There are two proofs given. Both are analytic. The first is a variant
of the traditional method using Mellin inversion of Dirichlet series
while the second is a proof due to Newman published in 1980. At this point
one could say ``mission accomplished'' and walk away quite satisfied that
one has a grasp of one of the truly great theorems in mathematics.
But Jameson does not leave us there.

The remaining three chapters can be read independently of each other. So
if this book is used as a text topics can be selected from these chapters
according to the time available and the particular interests of the
instructor and students. I will deal with chapter six first.

Chapter six again returns to the proof of Prime Number Theorem. The proofs in
Chapter three are analytic in the sense that they use complex analysis.
But using using integrals in the complex plane to prove results about
prime integers strikes most people as quite an unusual way to go about
it, though clearly it can be done. Chapter six is devoted to the
``elementary'' proof published by both Selberg and Erdos in 1948. In
particular it follows the method of Levinson. The material here is
quite advanced but does lead to a proof and the reader should feel a
sense of accomplishment if they successfully complete this chapter.

Chapter four is about prime numbers in residue classes and Dirichlet's
theorem while chapter five devotes itself to error estimates and
the Riemann Hypothesis. Again, this is interesting material and well
worth the effort required to understand it.

The bibliography is small and, I believe without exception, the references
are all books; there are no journal articles cited.

The index is small but I found it adequate.

Overall, the book has a most interesting topic and is very well
written. However, there is one black mark against the book which
prevents me from awarding all five stars. I expect most readers who
buy this book will not have the benefit of using it in an instructor lead
course, rather they will be studying it independently because accounts
of the prime number theorem and its proof are only infrequently encountered
in textbooks. The problem is that while the exercises are appropriate,
though perhaps there are too few of them, there is no answers to consult
if you get stuck. The material is not easy, each exercise takes quite a bit
of work and thinking. This is what you would expect at this level, but
the problem remains that there is nothing to aid a reader should they get stuck.
The author would do well to have a look at how Eymard and Lafon have
handed answers to exercises in their book on Pi.

Despite that one drawback I recommend this book both as a textbook for a course
and as a book for self-study.