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Elementary methods in number theory. (English) Zbl 0953.11002
Graduate Texts in Mathematics. 195. New York, NY: Springer. xviii, 513 p. DM 98.00; öS 716.00; sFr. 89.50; £34.00; \$ 49.95 (2000).

This book is more and less than an “Introduction to number theory”; less in the sense that some topics like algebraic numbers are completely excluded, and more as it includes results that are elementary, as the title says, but rather difficult.

The book is divided into three parts. Part I, “A first course”, starts indeed with the basics, divisibility and congruences up to quadratic reciprocity. Still, there are some remarkable innovations. One is the introduction of Fourier analysis at an early stage. Fourier transforms are considered in finite Abelian groups only; this saves the reader from the problems of measurability and convergence, while every important phenomenon is there, like Poisson summation and trace formulae. This is used to treat Gauss sums here, and easily provides the necessary properties of characters to study primes in arithmetic progressions later. It is also an excellent preparation for “harder” Fourier analysis (which is outside the scope of this book).

The final chapter of Part I tells us about the $abc$ conjecture, Mason’s theorem and the analog of Fermat’s last theorem for polynomials, beautiful results that deserve to be widely known.

The principal aim of Part II (Chapters 6-10), Multiplicative number theory, is to present the prime number theorem and Dirichlet’s theorem on primes in arithmetic progressions. Chapter 6 develops elementary convolution calculus. Chapter 7 tells some properties of the number and sum of divisors, including the existence of density for the set of abundand numbers, with Erdős’s proof. Chapter 8 gives some elementary estimates for primes, and as an application, Hardy and Ramanujan’s theorem for the normal number of prime divisors, with Turán’s proof.

Chapter 9 contains Selberg’s elementary proof of the prime number theorem. (Personally I would have selected Postnikov and Romanov’s version instead.)

Chapter 10 is devoted to Dirichlet’s theorem. There is an interesting novelty in the proof that $L\left(1,\chi \right)\ne 0$ for complex characters, which is probably the author’s (he does not claim credit for it, nor does he attribute it to anybody else; it was new to me). This consists in showing that ${\sum }_{n\le x}\chi \left(n\right){\Lambda }\left(n\right)/n=-logx+O\left(1\right)$ if $L\left(1,\chi \right)=0$, and then deduce that ${\sum }_{p\le x,p\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}m\right)}\left(logp\right)/p$ would be negative. The underlying idea is the same as in the usual analytic proof, but it is beautifully translated into another language.

Part III is devoted to some problems of additive number theory. Chapter 11 presents Linnik’s elementary solution to Waring’s problem. Chapter 12 gives a generalization, due to the author, which sounds as follows. If ${f}_{1},\cdots ,{f}_{s}$ are integer-valued polynomials of degree $k$ and leading coefficients in the interval $\left(0,c\right]$, then, for $s>{s}_{0}\left(k\right)$, the lower density of integers representable in the form ${f}_{1}\left({x}_{1}\right)+\cdots +{f}_{s}\left({x}_{s}\right)$ is at least $\delta \left(k,c\right)>0$.

Chapter 13 is a discovery in the other sense of the word: a miraculous and essentially forgotten formula of Liouville is revived. In this chapter it is applied to find the representation of primes by the forms ${x}^{2}+{y}^{2}$ and ${x}^{2}+2{y}^{2}$. During the proof we find a recurrence relation for $\sigma \left(n\right)$, the sum of the divisors (not exacly the familiar one). Further applications are given in Chapter 14, where representations by sums of squares are discussed.

Chapter 15 is devoted to the estimation of the partition function. The author stops at a logarithmic asymptotics ($logp\left(n\right)\sim {c}_{0}\sqrt{n}$). The final Chapter 16 considers generalizations to partitions with summands restricted to a subset $A$ of the positive integers. The main results say that if $A$ has asymptotic density $\alpha >0$, then the corresponding partition function satisfies $log{p}_{A}\left(n\right)\sim {c}_{0}\sqrt{\alpha n}$, and conversely, if ${p}_{A}$ admits such an asymptotics, then $A$ must have density $\alpha$.

This is a nice book, with carefully selected beautiful theorems. The proofs are meticulously explained in every detail (in fact, the presentation is somewhat too detailed for my taste). There are historical remarks and numerous exercises, mostly on the easy side. The book is not intended for a general course in number theory, but it is possible to base more specialised courses on it at the introductory or medium level. Besides there are many ideas that can be incorporated into a course which follows a different path. Some material, especially in Part III, is valuable even to experts.

MSC:
 11-01 Textbooks (number theory) 11Axx Elementary number theory 11Nxx Multiplicative number theory 11Pxx Additive number theory; partitions 11Mxx Analytic theory of zeta and $L$-functions 11Dxx Diophantine equations