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Introduction to Analytic Number Theory

K. Chandrasekharan
Publisher: 
Springer-Verlag
Publication Date: 
1969
Format: 
Hardcover
Series: 
Grundlehren der Mathematischen Wissenschaften 148
Price: 
109.00
ISBN: 
978-3-642-46126-2
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
08/4/2017
]

This is a concise sampler of analytic number theory, introducing problems in a number of different areas of number theory and showing how they can be attacked by analytic methods. It doesn’t go into a great deal of depth on any problem. The prerequisites are a moderate knowledge of complex analysis. The necessary number theory is developed from scratch. In some ways it resembles D. J. Newman’s Analytic Number Theory, although that book deals with more difficult problems and is not really an introduction.

The exposition owes a great deal to Hardy & Wright’s An Introduction to the Theory of Numbers and to Landau’s Elementary Number Theory, except for the last two chapters, on Dirichlet’s theorem on primes in arithmetic progressions and on the Prime Number Theorem. The former uses a complex-variables proof depending on Landau’s theorem that a positive-term Dirichlet series has a singularly on the abscissa of convergence. The latter is a nice exposition of the Wiener-Ikehara proof that requires no knowledge of Fourier integrals. Other topics covered include Bertrand’s postulate, quadratic reciprocity (through Gauss sums), average orders of arithmetic functions, equidistribution, and a good bit on the geometry of numbers.

The big weakness of this book, apart from its high price, is that it’s too shallow and doesn’t lead you anywhere. For example, it does have a very nice proof of the prime number theorem, but you don’t find out that there is a whole industry devoted to improving our knowledge of the Riemann zeta function or how it connects to prime number theory. A good alternative to this book is Apostol’s Introduction to Analytic Number Theory. It also starts at the beginning, covers the roughly the same topics, but goes into much greater depth. Apostol also has very good exercises, while the present book has none.


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.