A Primer of Analytic Number Theory: From Pythagoras to Riemann 1st Edition by Jeffrey Stopple (PDF)

Description

This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. Jeffrey Stopple pays special attention to the rich history of the subject and ancient questions on polygonal numbers, perfect numbers and amicable pairs, as well as to the important open problems. The culmination of the book is a brief presentation of the Riemann zeta function, which determines the distribution of prime numbers, and of the significance of the Riemann Hypothesis.

User’s Reviews

Editorial Reviews: Review “The book is interesting and, for a mathematics text, lively…. Stopple has done a particularly nice job with illustrations and tables that support the discussions in the chapters.” Chris Christensen, School Science and Mathematics”… this is a well-written book at the level of senior undergraduates.” SIAM Review”The book constitutes an excellent undergraduate introduction to classical analytical number theory. The author develops the subject from the very beginning in an extremely good and readable style. Although a wide variety of topics are presented in the book, the author has successfully placed a rich historical background to each of the discussed themes, which makes the text very lively … the text contains a rich supplement of exercises, brief sketches of more advanced ideas and extensive graphical support. The book can be recommended as a very good first introductory reading for all those who are seriously interested in analytical number theory.” EMS Newsletter”… a very readable account.” Mathematika”The general style is user-friendly and interactive … a well presented and stimulating informal introduction to a wide range of topics …” Proceedings of the Edinburgh Mathematical Society Book Description An undergraduate-level 2003 introduction whose only prerequisite is a standard calculus course.

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

⭐This is a great book. It presents the material in a completely rigorous fashion, yet is always providing an intuitive description and a great deal of motivation. This is a real plus and a refreshing break from the usual theorem/proof litany of typical math books (of course there are some exceptions which are likewise outstanding). The pace of presentation is conducive to really absorbing the material, and all along the way, there are problems that clarify and enhance your grasp of what is presented. Generously, there are solutions in the back, so, if all else fails, you can resort to them and continue making progress.EDIT: The more time I spend in this book, the more I like it. IT really is a gift in that it teaches you how to think in terms of analytic number theory. Too often texts are either too pithy or seem to think you are already familiar with the subject. Thus it is a rare find to have a text that is fully rigorous,but opens your eyes to what’s really going on – so there is little tendency to miss subtle points or have aweak base of understanding.Another feature of this book is that it is truly self-contained. There are sections devoted to background material (series, complex numbers, etc.) which are directly pertinent to the subsequent presentation. Even if you are familial with this material, the author has a knack for pointing out ways of looking at things that are quite relevant and that you might not have thought of.Whether for a course or self-study, as a fellow member of a great math online website said: this book is a pleasure.Regarding the KINDLE EDITION: as Bourbaki might say: WARNING – this book does not reproduce well. The notation is sometimes split from the end of one line to the beginning of the next without maintaining correct “grammar.” Also maneuvering to prior equations when referred to in the text, and going back and forth from the solutions to the text takes some doing.But the print editions is very nicely formatted and visually accessible. I personally feel this is not a trivial point since it lends itself to easy focus and the retention of equations as visual images.

⭐This book is a wonderful introduction to analytic number theory at the sophomore level. It assumes single-variable calculus but little beyond and covers the standard topics as well as introducing some topics on the edge such as the Birch and Swinnerton-Dyer conjecture.The Kindle edition uses the mobi format which is executed a bit better in this book than in most such books with heavy mathematical content. There are, however, consistent indenting errors and the like which detract from the readability of the text versus a print replica presentation, and the figures are inserted as graphics which do not resize with changes in font size and which have a white background that defeats the use of the sepia and black backgrounds in Kindle eReaders. The use of the mobi format as executed for this text does it a disservice considering the excellence of the content and typesetting appearance in physical print.

⭐This is definitely one of the delightful math books to read. The material is so well organized that it flows very nicely. This book provides a very gentle introduction to such topics as Zeta function and Prime Number Theory.

⭐Awesome. A great place to start learning ANT.

⭐A good start before tackling Apostol’s fundamental “Introduction to Analytical NumberTheory”.Original, clear, didactic approach… reminding of Derbyshire’s style but in a more formal manner.On the negative side : bad numbering of sections, theorems, chapters ; general layout and architecture of sections and paragraph, intra-chapter and inter-chapter logical patterns, leaving something to be desired ; many exercises in lieu of formal text, leading to unfortunate fragmentation…

⭐There usually seems to be a pretty big gap between the background needed to understand books on elementary number theory and what’s needed to understand most books on analytic number theory, and this book does a good job of making that gap seem smaller. The writing feels a bit like Silverman’s “Friendly Introduction to Number Theory” and Derbyshire’s “Prime Obsession.” There are plenty of experiments for Mathematica and Maple. I could see this book being used in an undergraduate number theory class. It doesn’t assume any familiarity with complex variables. If you can integrate and aren’t too afraid of series or logarithms, this book should be no problem.The book goes over multiplicative functions, Mobius inversion, the Prime Number Theorem, Bernoulli numbers, the Riemann zeta function (value at 2n, analytic continuation, functional equation, the Riemann Hypothesis), the Gamma function, Pell’s equation, quadratic reciprocity, Dirichlet L-functions, elliptic curves (and EC L-functions and the BSD conjecture), binary quadratic forms, and an analytic class number formula for imaginary quadratic fields.I recommend this book to anyone who can read. And for those who can’t read, this book is good motivation to become literate.

⭐A little background on me. I have just finished my freshman year of high school, and this was my first book on number theory. However, I have read many other math texts. In the beginning of the book there are some new concepts introduced, but they are not too hard to understand. The middle is refreshing as it involves a lot of calculus, which the student is most likely familiar with. The latter part consists of a variety of new ideas, and the theorems can get quite lengthy. I do not fully understand all of them myself. The book is well written and also includes the history of many mathematical problems.

⭐In preparation for M823 Analytic Number Theory (OU) I’d purchased Apostol and found it very terse and dry. I purchased this book as an alternative and have so far found it absolutely fantastic! The exercises build confidence, insight and intuition. His explanation of Big Oh notation work well for me as has the refresher on Taylor series.The brief history backgrounds are interesting and help to add further colour to the book.The author clearly teaches undergraduates and knows what mistakes are likely to be made, and I’ve made a few of them.I only hope I can finish the book before the course starts in October.

⭐excellent purchase and a good introduction to analytic number theory.

⭐This is a textbook like textbooks should be. Everything is explained extremely simple (while the results are not alwyas simple) and with patience. I especially like the appraoch where you have to guess theorems, come up with your own conjectures and try to prove them. Unlike other books that simulate this approach, this book contains a comprehensive explanation of the exercices. The important theorems that you were supposed to come up with yourself (and this is possible) are later repeated and prooved in the text (or in the extensive exercise hints).This simplicity doesn’t keep the author from reaching limits as high as Riemann so let’s all take him as an example.

⭐A good start before tackling Apostol’s fundamental “Introduction to Analytical NumberTheory”.Original, clear, didactic approach… reminding of Derbyshire’s style but in a more formal manner.On the negative side : bad numbering of sections, theorems, chapters ; general layout and architecture of sections and paragraph, intra-chapter and inter-chapter logical patterns, leaving something to be desired ; many exercises in lieu of formal text, leading to unfortunate fragmentation…

⭐Yet to be read but first glance worth having.

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