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Representations of integers as sums of squares. (English) Zbl 0574.10045
New York etc.: Springer-Verlag. XI, 251 p. DM 148.00 (1985).

This book arose from lectures delivered by the author at the Israeli Institute of Technology in Haifa during the academic year 1980-1981. The main idea of the book is to make the arithmetic theory of quadratic forms (especially sums of squares) accessible to a wide circle of readers, while this theory helps to solve not only some problems in mathematics, but also has direct applications in crystallography, electrostatics, potentials of charge distributions, some boundary value problems in quantum physics and classical mechanics. Most of the material of the book is classical, however some proofs of these old results are new; there are also results belonging to the author himself.

The book consists of fourteen chapters. The first is preliminary. The following three chapters are devoted to criteria of representability of an integer n as a sum of 2,3 and 4 integer squares; to formulae for the arithmetical function ${r}_{k}\left(n\right)$, the number of representations of n as a sum of k integer squares, when $k=2,3,4$; to the number of lattice points in the circle and in the sphere.

Chapter 5 deals with Legendre’s theorem on the diophantine equation $a{x}^{2}+b{y}^{2}+c{z}^{2}=0$. In chapter 6 some problems of representations of n by exactly k nonvanishing squares is discussed. Chapter 7 is devoted to the problem of finding the number of essentially distinct representations of n as a sum of squares.

In chapter 8 the theory of Jacobian theta-functions is sketched. In chapter 9 by means of theta-function and Lambert series formulae for ${r}_{k}\left(n\right)$ are derived and estimates of ${r}_{k}\left(n\right)$ are obtained with k even and $2\le k\le 12$. Chapter 10 is devoted to mention without proofs various recent results on the problem of representations by sums of squares and related topics.

Chapter 11 deals with some well-known facts about Farey series, Gaussian sums, of the basic theory on modular forms and functions. In chapter 12 the principle of the circle method is exposed; then Hardy’s asymptotic formula for ${r}_{k}\left(n\right)$, $k\ge 5$ and the explicit evaluation of the singular series of the problem are given. In chapter 13 two methods for evaluating the function ${r}_{k}\left(n\right)$ are sketched: Estermann’s method, using only elements of the theory of functions of complex variables, and the method of modular functions. Chapter 14, with only the barest sketches of proofs, lists some recent developments, in particular on the representation problem for integers in algebraic number fields and that of positive definite functions by squares and more general quadratic forms.

For the illustration of some exposed results a large number of examples are given. Some results are stated without proofs. Each chapter is supplemented by exercises, historical comments, formulations of more advanced and mentioning of recent results. The book also contains an appendix of 6 open problems and concludes with references and an extensive bibliography of more than 550 items from the period 1750-1984. For the literature before the eighteenth century, the reader is often directed to L. E. Dickson’s ”History of the theory of numbers”. The book ends with author and subject indexes.