## A primer of analytic number theory. From Pythagoras to Riemann.(English)Zbl 1029.11001

Cambridge: Cambridge University Press. xiii, 383 p. (2003).
This is a wide ranging introduction to analytic number theory. Despite the title, the book is concerned only with that part of the subject which concerns the Riemann zeta-function and its relatives. There is no mention of additive number theory, sieves, or exponential sums, for example. Historically the reader is taken from the Rhind papyrus through to Wiles’ proof of Fermat’s Last Theorem. Mathematically the range is from the summation of finite arithmetic progressions to the functional equation for the Riemann zeta-function. The book contains a fair bit of basic analysis, with discussions of the fundamental theorem of calculus, and of the convergence of series, for example. Within number theory the book considers such issues as perfect and amicable numbers, Dirichlet’s divisor problem, Pell’s equation, and elliptic curves and their $$L$$-functions.
Inevitably the pace is rather fast in places, and the proofs of key results – the law of quadratic reciprocity, for example – are often omitted. Indeed the author takes the brave step, for a book on analytic number theory, of leaving out a problem of the prime number theorem! By way of compensation there is a lot of interesting historical and motivational material, and this is the real strength of this book. It is perhaps best suited as introductory reading on number theory in general, leading the reader to the study of zeta-functions, rather than being a formal textbook with detailed proofs. It would be excellent background reading for undergraduates at any stage of their course.

### MSC:

 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 11Mxx Zeta and $$L$$-functions: analytic theory 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$

### Keywords:

textbook; analytic number theory; Riemann zeta-function