zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 (n), 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 ax 2 +by 2 +cz 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 (n) are derived and estimates of r k (n) are obtained with k even and 2k12. 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 (n), k5 and the explicit evaluation of the singular series of the problem are given. In chapter 13 two methods for evaluating the function r k (n) 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.

Reviewer: G.Lomadze

MSC:
11-01Textbooks (number theory)
11P05Waring’s problem and variants
11E16General binary quadratic forms
11P55Applications of the Hardy-Littlewood method
11D09Quadratic and bilinear diophantine equations
11D85Representation problems of integers
11E45Analytic theory of forms