Multiplicative Number Theory (Graduate Texts in Mathematics, 74) 3rd Edition by Harold Davenport (PDF)

Description

The new edition of this thorough examination of the distribution of prime numbers in arithmetic progressions offers many revisions and corrections as well as a new section recounting recent works in the field. The book covers many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetical progressions and the theorem of Siegel. It also presents a simplified, improved version of the large sieve method.

User’s Reviews

Editorial Reviews: Review From the reviews of the third edition:”The book under review is one of the most important references in the multiplicative number theory, as its title mentions exactly. … Davenport’s book covers most of the important topics in the theory of distribution of primes and leads the reader to serious research topics … . is very well written. … is useful for graduate students, researchers and for professors. It is a very good text source specially for graduate levels, but even is fruitful for undergraduates.” (Mehdi Hassani, MathDL, July, 2008)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I recommend against using this as an introduction to analytic number theory. It is indeed elegantly written, but many of the calculations are abbreviated too much for a beginner to follow. For example, it took me 13 pages to write out sections 24 and 25 on Vinogradov’s estimate for exponential sums over primes in detail, still taking some things for granted. It is written as a sequence of milestones rather than a full itinerary. A much more transparent presentation of Vinogradov’s estimate and the Hardy-Littlewood circle method is given in Nathanson,

⭐, Chapters 5 and 8.The book is organized into short sections and results are usually not labelled and often it is not even mentioned when a result like Merten’s theorem is used. But this is a good book. The writing is clear although almost always too brief, and strong results are proved. This is a certainly a pleasant book to browse once you know the subject, and it has the feel of lectures and might be useful for writing lecture notes. Almost all the results in Davenport are proved in Montgomery and Vaughan,

⭐, which gives many more details of calculations and easy to navigate. If you want an introduction to analytic number you, I strongly recommend Montgomery and Vaughan. It is indeed much longer than Davenport, but this is because it has more material and also gives much longer proofs, which are in fact easier to read because they move with steps that a learner can follow.

⭐I am a Analytical Number Theory math major, and this book was recommended to me.

⭐H. Davenport is Littlewood’s student. This book is very classical. It introduces zeta functional equation and Dirichlet theorem via a historical view.

⭐Work through this book. While Serre’s

⭐is slicker, it is nowhere near as enlightening. Iwaniec’s treatise

⭐is a good reference for professionals, but unreadable for someone who has not seen (a lot of) the material before. Davenport’s book is very clear and very deep at the same time. The recent editions of this book have been brought up to date, but the core has not changed too much, so don’t feel obligated to buy the latest edition.

⭐Ever since I first read about the prime number theorem, I have been roaming the mathetmatical landscape, looking for the best proof of this result. I believe this book has it. It’s not the simplest or the shortest proof, but it gives the deepest understanding of why the prime numbers behve like they do. In addition to this, it shows you the historical perspective in these proofs. All too often today math books give one short and slick proofs that leave you wondering how on earth they came up with it. In this book, however, one can almost feel the thoughts going through Riemann and Dirichlet’s heads as they came up with the theorems. This book also has the proof of Dirichlet’s theorem and Vinogradov’s partial proof of the ternary goldbach conjecture. The vinogradov and following sections are considerably harder, partly because they were not written by Davenport himself. Anyway, if you’re serious about Analytic number theory and how mathematicians think, this books needs to be on your bookshelf.

⭐very good book to study analytic number theory

⭐解析学（複素解析、実解析、フーリエ解析など）を用いて数の性質を研究する分野を「解析的数論」と言い、本書はこの理論の面白さを教えてくれる好適な入門書である。本書の魅力・特徴として、以下の３点を挙げたい。先ず、解析的数論の歴史的な発展に沿って記述されている内容と構成が素晴らしい。算術級数の素数定理の証明にディリクレが導入したL関数が２次体や２次形式の類数公式に自然に結びつくことが解説され（６章まで）、素数分布の解明と言う問題意識から導入されたリーマンζ関数やL関数の基本的な性質（関数等式、部分分数展開、零点を持たない領域、零点分布の漸近式、明示公式、素数定理の証明への応用、など）が詳しく解説され（22章まで）、更に後半の７つの章では三角和と大きな篩の解説からゴールドバッハ-ヴィノグラドフの定理とボンビエリの平均値定理という近代的解析数論の頂点に位置する成果の解説にまで及んでいる。これだけ充実した内容が僅か174頁の中に盛り込まれているのである。次に、ディリクレのL関数の基本的な性質が、リーマンζ関数のそれと対比する形で述べられており、分かり易く非常に参考になる。複素解析を用いて解析数論の深い結果を得るには、ζ関数に加えL関数の詳しい性質が必要になるが、ここまで詳しくL関数の性質が論じられているのは非常に有益である。最後に、著名な定理に簡易化された証明が述べられており、とても教育的である。L関数の例外零点に関するジーゲルの定理へのエスターマンによる証明や三角和の評価及びボンビエリの定理へのヴォーガンによる新証明などはこの例である。本書に関連する話題を詳しく扱うザギヤー著『数論入門』、三井著『整数論』、ヴィノグラドフの三角和（Trigonometrical Sums）の教科書などを参照されると更に理解が深まるので、併読される事をお薦めしたい。

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