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The development of prime number theory. From Euclid to Hardy and Littlewood. (English) Zbl 0942.11002
Springer Monographs in Mathematics. Berlin: Springer. xii, 448 p. DM 169.00; öS 1234.00; sFr. 153.00; £58.50; \$ 94.00 (2000).

This is a most welcome addition to the literature on prime numbers, zeta and $L$-functions and arithmetical functions. As the subtitle “From Euclid to Hardy and Littlewood” says, the author has restricted himself chronologically in giving the historical development of prime number theory and related topics from the very beginnings until the first decades of the XXth century. Nevertheless in the notes accompanying the text the reader may find much material on most recent developments involving the subjects treated in this book. The book is divided into six chapters:

1. Early Times. 2. Dirichlet’s Theorem on Primes in Arithmetic Progressions.

3. Chebyshev’s Theorem. 4. Riemann’s Zeta-function and Dirichlet series. 5. The Prime Number Theorem. 6. The Turn of the Century.

This is followed by References (the most extensive this reviewer has ever seen in any work, containing almost eighty pages!), Author index and Subject index. The material in each chapter is given historically, as a series of theorems, some of which are proved in various ways.

The book starts with Theorem 1.1: “There are infinitely many prime numbers” and ends with the famous conjectures of Hardy and Littlewood on the asymptotic formulas for the representation of an odd integer as prime and a double of a square etc.

The style is clear, with just the right amount of details. Each chapter closes with carefully chosen Exercises. Novices and experts alike will find that this a book of highest quality, which sets a standard for future works dealing with the history of Mathematics.

##### MSC:
 11-02 Research monographs (number theory) 11-03 Historical (number theory) 11A41 Elementary prime number theory 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11N05 Distribution of primes 11N13 Primes in progressions 11N25 Distribution of integers with specified multiplicative constraints 11M26 Nonreal zeros of $\zeta \left(s\right)$ and $L\left(s,\chi \right)$; Riemann and other hypotheses