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Zbl 0942.11002
Narkiewicz, Władysław
The development of prime number theory. From Euclid to Hardy and Littlewood. (English)
[B] Springer Monographs in Mathematics. Berlin: Springer. xii, 448 p. DM 169.00; öS 1234.00; sFr. 153.00; \sterling 58.50; \$ 94.00 (2000). ISBN 3-540-66289-8/hbk

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: \par 1. Early Times. 2. Dirichlet's Theorem on Primes in Arithmetic Progressions. \par 3. Chebyshev's Theorem. 4. Riemann's Zeta-function and Dirichlet series. 5. The Prime Number Theorem. 6. The Turn of the Century.\par 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. \par 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. \par 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.
[A.Ivić (Beograd)]

MSC 2000:: *11-02 Research monographs (number theory)
11-03 Historical (number theory)
11A41 Elementary prime number theory
11M06 Riemannian zeta-function and Dirichlet L-function
11N05 Distribution of primes
11N13 Primes in progressions
11N25 Distribution of integers with specified multiplicative constraints
11M26 Nonreal zeros of zeta(s) and L(s,chi)

Keywords: prime numbers; Riemann zeta-function; arithmetic functions; exercises

Cited in: Zbl 1196.11004 Zbl 1177.01053 Zbl 1028.11050

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