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Divisors. (English) Zbl 0653.10001
Cambridge Tracts in Mathematics, 90. Cambridge (UK) etc.: Cambridge University Press. xvi, 167 p. £25.00; $ 39.50 (1988).
The aim of these authors is to present a cohesive study, with a logical structure, of questions concerning the divisors of an integer, the main focus being on divisors in short intervals and the propinquity of divisors, and this goal has been successfully achieved. The modern theory has its roots in work of Hardy and Ramanujan and owes a great deal to the perceptiveness of Paul Erdős, whose continuing influence on the subject is very evident here and elsewhere. In the last decade, much progress had been made in developing our understanding of the behaviour of divisors, and the present authors have been in the forefront of this sphere of activity. Many of the results established in this book have been available previously only in the research journals, and some proofs are given here in detail for the first time.
This volume is likely to be an important reference book in an active research field for a considerable time. It is intended for graduate students in analytic number theory and others interested in gaining an insight into this rich area either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style.
For the reader new to the field (and for others!), this book is demanding and the proofs (inevitably) are very technical, but the authors take care to give sufficient explanation and motivation to help those prepared to persevere to follow the arguments and to gain a familiarity with the methods used. These incorporate well known techniques from Analysis and Number Theory, but a probabilistic train of thought is also apparent. The diligent reader should be proficient in the use of Hölder’s inequality by the time he reaches the end of the book!
Each chapter concentrates on one main theme, which is clearly set in context, and ends with some useful background notes and a wealth of challenging exercises. A collection of results required subsequently are given in chapter 0, whilst the objective of chapter 1 is to obtain optimal results concerning the size, for almost all positive integers \(n\), of \(|\omega(n,t)-\log\log t|\), where \(\omega(n,t) = \text{card}\, \{p: p \mid n,\;p \leq t\}\) (\(p\) prime). Chapters 2 and 3 are devoted, respectively, to estimating the function \(H(x,y,z)\), the number of positive integers \(n \leq x\) with a divisor \(d\) satisfying \(y < d \leq z\), and to examining the function \(\tau(n,\theta) = \sum _{d\mid n} d^{i\theta}\) (\(\theta\) real); this latter function was first systematically studied in this way by the first author [J. Lond. Math. Soc., II. Ser. 9, 571–580 (1975; Zbl 0308.10037)].
In the next chapter, functions measuring in some way the propinquity of the divisors of \(n\) are investigated, examples being the function \[ T(n,\alpha) = \text{card}\, \{d,d': d\mid n,\;d'\mid n,\;| \log d / d'| \leq \log^\alpha n\} \text{ for } \alpha \leq 1 \] and Erdős’ function \[ \tau^+(n) = \text{card}\, \{k \in Z : \exists d\mid n: 2^ k < d \leq 2^{k + 1}\}. \]

The motivation for chapter 5 is Erdős’ conjecture, made nearly half a century ago, that almost all integers have two divisors \(d,d'\) satisfying \(d<d'\leq 2d\); this was finally proved, in a stronger form, by H. Maier and the second author [Invent. Math. 76, 121–128 (1984; Zbl 0536.10039)]. The function \[ \Delta_ r(n) = \max _{u_ 1,...,u_{r- 1}}\text{card}\,\{d_ 1...d_{r-1}\mid n: u_ i < \log d_ i \leq u_ i + 1\quad (1 \leq i < r)\} \] was first introduced by C. Hooley [Proc. Lond. Math. Soc., III. Ser. 38, 115–151 (1979; Zbl 0394.10027)] in a wide ranging paper that highlights the potential in applications for results of the type described in this volume; the main task of chapters 6 and 7 is to establish good bounds for the sum \[ \sum_{n\leq x} \Delta_ r(n) y^{\omega(n)}\quad (y > 0) \] when \(y\) lies outside and within a certain critical interval.
At the end of the book, there is a comprehensive and useful list of references. A combination of the notation section at the beginning and the index at the end will assist in the task of locating key definitions and results. It might perhaps have been helpful to the reader from another area of mathematics if the authors had included in the notation section an explanation of the few symbols that might be unfamiliar to someone from outside number theory, but this is a very minor point.
Altogether, this volume is a very valuable addition to the stock of current, research level books and makes a major and worthwhile contribution to the literature.

MSC:
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11N05 Distribution of primes
11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions in probabilistic number theory
11B83 Special sequences and polynomials
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