The book under review is fitted with a back-cover blurb which is exquisitely tantalizing:
Arithmetic Tales offers a comprehensive introduction to these topics … with an emphasis on analytic number theory. Since it requires very little technical expertise it will appeal to a wide target group including upper level undergraduates, doctoral and masters students.
Of course this leads one immediately to ask for specifics: what are “these topics” and are they as accessible as is claimed? To answer the second question first, the answer is a guarded “yes.” The material is deep and central to number theory and, given the nature of the field, is locally accessible, so to speak: typically one can get to research level questions in “the higher arithmetic” (to borrow Davenport’s phrase) by becoming a jack of many trades (“working locally”), as opposed to a master of any (let alone many) of them. A number theorist needs to know a lot of complex analysis, for instance, and Fourier analysis, and sundry forms of algebra, but it’s not necessary to seek expertise at the highest level, at least not just out of the gate. You don’t need to know everything there is to know about Riemann surfaces, for instance, to learn automorphic forms. However, you do need to know something; just be prudent in how you go about it lest you spread yourself too thin. A good advisor will know how to go about guiding a rookie in this art of reading and studying preparatory to going at research with guns blazing. Number theory is a particularly apposite case in point: the game is found early in the expedition, when the hunter is just getting the hang of the arsenal.
This takes us to the game presented by Bordellès in his text of more than 500 pages, seven chapters, and an appendix. The latter, by the way, testifies to Bodellès’ pedagogical focus in writing the book: the appendix consists of seven sets of “hints and answers to exercises” in one-to-one correspondence to the foregoing chapters. The chapters in turn provide the answer to the first question asked above, namely, the nature of the “topics” that are advertised on the back cover. Indeed, these include
Mertens’ theorem … Chebyshev’s inequalities and the celebrated prime number theorem [giving] estimates for the distribution of prime numbers … Dirichlet’s convolution product which arises with the inclusion of several summation techniques and a survey of classical results such as Hall and Tenenbaum’s theorem and … Möbius inversion …
Beyond this we encounter
counting of integer points close to smooth curves and its relation to the distribution of squarefree numbers … exponential sums and algebraic number fields.
Thus, in a sense, Bodellès proceeds from elementary and analytic number theory to algebraic number theory — something of an inversion, these days, but a very sensible way to proceed. The first and second chapters, for instance, present a sequence of themes and methods that, while gorgeous and important, are in many ways all but self-contained and require very little in the way of a student’s preparation; an exemplar of this elementary material is what Bordellès calls the Bachet-Bézout theorem, viz. that if gcd(a,b) = 1 then for some u, v we get ua + vb = 1, and conversely; cf. p.27 ff. So, it really is a slow and comfortable start. But by the third chapter number theory proper is under way and under the heading “Prime Numbers,” we go from the fundamental theorem of arithmetic to Chebyshev, the Riemann ζ-function, primes in arithmetic progressions, and sieve methods. By the end of the third chapter, after some 160 pages, we are swimming in deep waters.
Bordellès then proceeds with chapters on “arithmetic functions,” “integer points close to smooth curves,” and “exponential sums,” including material on Selberg’s sieve, the method of Filaseta and Trifonov (very sophisticated stuff, really — see p. 281 ff.), and (qua exponential sums) material by Van der Corput, Hardy-Littlewood (well, it’s really the discrete method, due to Bombieri, Huxley, Iwaniec, Mozzochi, and Watt; see p.v), and Vinogradov. As already hinted, the book ends with a chapter on “algebraic number fields” and, here, too, Bordellès starts “nice and easy” and ends “nice and rough” (to quote Ike and Tina Turner): he goes from rings, fields, modules, and Galois theory (in the broad sense), through the multiplicative side of ideal theory (in number fields), to Brauer-Siegel, Kronecker-Weber, class field theory, and more.
The book under review should succeed very well as a source from which to learn a lot of very beautiful number theory in an accessible way (modulo a good output of energy on the part of the reader). This book is clearly a true labor of love and Bordellès has produced an important text and an elegant scholarly work.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.