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Handbook of number theory. (English) Zbl 0862.11001

Mathematics and its Applications (Dordrecht). 351. Dordrecht: Kluwer Academic Publishers. xxvi, 622 p. (1995).
From the Preface: “It is the aim of this book to systematize and to present in an easily accessible framework the most important parts of number theory, which are expressed by inequalities or by estimates.” So, this monograph helps students and teachers of number theory, and researchers in this field to get thoroughly informed on many topics in elementary, analytic and probabilistic number theory. The authors describe important results, beginning with the “results of the pioneers in the domains regarded”, and they give “some results reflecting the evolution from the pioneer works up to recent ones”. Moreover, a lot of bibliographic references are given. The wealth of the material covered in this monograph may be guessed from a short description of the contents.
I. Euler’s \(\varphi\)-function (elementary problems, asymptotic formulae for sums containing the values \(\varphi(n)\), iterates of \(\varphi,\dots\)).
II. The divisor function, its analogues and generalizations (maximal order of \(d(n)\), highly composite numbers, \(\sum_{n\leq x}d(n)\), divisor problem in arithmetic progressions, iteration of \(d(n),\dots\)).
III. The sum of divisors, perfect numbers.
IV. Sums over \(p_{\max}(n)\), \(\log p_{\max}(n)\), greatest prime factors of polynomials, integers without large prime factors, Dickman’s function, \(\dots\).
V. \(\omega(n)\), \(\Omega(n)\) and related functions.
VI. The Möbius function, \(k\)-free and \(k\)-full numbers.
VII., VIII. Prime numbers (prime number theorem, primes in short intervals, difference between consecutive primes, Chebyshev’s conjecture, \(\dots\)) and primes in arithmetical progressions, including Siegel’s theorem, the Brun-Titchmarsh theorem, and the Bombieri-Vinogradov theorem. {Reviewer’s Remark: In spite of the authors’ statement “the choice of subjects reflects the personality of the authors” the reviewer would have liked to find a chapter dealing with the analytic theory of the zeta-function and \(L\)-functions (functional equation, zeros, density theorems, \(\dots\)).}
IX. Additive prime number theory.
X., XI. Exponential sums and character sums (including large-sieve inequalities).
XII. Binomial coefficients, consecutive integers.
XIII. Estimates involving finite groups and semi-simple rings (for example: maximal order and mean-value of \(a(n)\), the number of non-isomorphic abelian groups of order \(n\), iterates of \(a(n)\), probabilistic results).
XIV. Partitions (asymptotic formulae, estimates, subsums of a partition, statistical theory of partitions, \(\dots\)).
XV. Congruences, residues and primitive roots (including Artin’s conjecture, least quadratic residues, least primitive roots, number of solutions of polynomial congruences, \(\dots)\).
XVI. Additive and multiplicative functions (including the Turán-Kubilius inequality, the Erdös-Kac and Erdös-Wintner theorem, local theorems for additive functions, almost periodicity, the Delange, Wirsing, Halász and Indlekofer theorems, Ramanujan expansions, \(\dots)\).
So this book contains much more material than the valuable forerunner by B. Spearman and K. S. Williams [Handbook of estimates in the theory of numbers (1975; Zbl 0393.10001)]. The reviewer thinks this monograph to be a great help to number theorists, and he thinks that all these will be grateful to the authors for doing the painstaking efforts of collecting and well-ordering an enormous mass of results. Unfortunately, the rather high price of this important publication might possibly prevent some people, highly interested in this monograph, from purchasing it.

MSC:

11-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to number theory
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11A41 Primes
11A07 Congruences; primitive roots; residue systems
11L03 Trigonometric and exponential sums (general theory)
11N05 Distribution of primes
11N13 Primes in congruence classes
11N37 Asymptotic results on arithmetic functions
11P32 Goldbach-type theorems; other additive questions involving primes
11P82 Analytic theory of partitions
11P83 Partitions; congruences and congruential restrictions

Citations:

Zbl 0393.10001
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