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.
A Primer of Analytic Number Theory
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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
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
This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The author 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 (some of which have million dollar prizes). The capstone 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.
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8 of 8 people found the following review helpful:
5.0 out of 5 stars
For the senior math undergraduate,
This review is from: A Primer of Analytic Number Theory: From Pythagoras to Riemann (Paperback)
A great book for senior undergraduates in mathematics or anyone with some background in calculus and complex numbers. Proofs are at a level where a careful reading makes them clear, and the author tells the reader when he is not being rigorous. Historical background and logical development of topics makes this a good read too. Most surprising to me was how the author tied in topics from prior chapters into later chapters--he didn't just jump from one topic to the next willy-nilly, but made the book flow as a whole. Problems given to the reader were helpful though sometimes too hard for me, a math major.
10 of 12 people found the following review helpful:
5.0 out of 5 stars
Do you like primers? ...and number theory? Well here you go!,
By
This review is from: A Primer of Analytic Number Theory: From Pythagoras to Riemann (Paperback)
There usually seems to be a pretty big gap between the math background needed to understand books on elementary number theory and what's needed to understand most books dealing with 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. The book 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 (and its value at 2n, its analytic continuation, its functional equation, and the Riemann Hypothesis), the Gamma function, Pell's equation, quadratic reciprocity, Dirichlet L-functions, elliptic curves (including their L-functions and the Birch and Swinnerton-Dyer 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.
6 of 7 people found the following review helpful:
5.0 out of 5 stars
One of my favorite math books,
By
This review is from: A Primer of Analytic Number Theory: From Pythagoras to Riemann (Paperback)
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.
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Inside This Book
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First Sentence:
The Greek word gnomon means the pointer on a sundial, and also a carpenter's square or L-shaped bar. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
kth gnomon, amicable pairs, polygonal numbers, congruent number, negative odd integer, nontrivial zeros, harmonic series diverges, binary quadratic forms, cattle problem, square modulo, harmonic numbers, triangular numbers, critical strip, quadratic reciprocity, series for exp, prime number theorem, logarithmic integral, zeta function Key Phrases - Capitalized Phrases (CAPs): (learn more)
Sophie Germain, Fundamental Theorem of Calculus, Abel's Theorem, Von Mangoldt, Analytic Class Number Formula, Thabit ibn Qurra, Second-order Harmonic, Chinese Remainder Theorem, Dirichlet L-functions, Hensel's Lemma, Birch Swinnerton-Dyer, Dictionary of Scientific Biography, Proof of Theorem, Leonardo of Pisa, Nicomachus of Gerasa, Bachet's Diophantus, Fundamental Theorem of Finite Calculus, Mean Value Theorem, Michael Psellus, Nicole Oresme New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The Greek word gnomon means the pointer on a sundial, and also a carpenter's square or L-shaped bar. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
kth gnomon, amicable pairs, polygonal numbers, congruent number, negative odd integer, nontrivial zeros, harmonic series diverges, binary quadratic forms, cattle problem, square modulo, harmonic numbers, triangular numbers, critical strip, quadratic reciprocity, series for exp, prime number theorem, logarithmic integral, zeta function Key Phrases - Capitalized Phrases (CAPs): (learn more)
Sophie Germain, Fundamental Theorem of Calculus, Abel's Theorem, Von Mangoldt, Analytic Class Number Formula, Thabit ibn Qurra, Second-order Harmonic, Chinese Remainder Theorem, Dirichlet L-functions, Hensel's Lemma, Birch Swinnerton-Dyer, Dictionary of Scientific Biography, Proof of Theorem, Leonardo of Pisa, Nicomachus of Gerasa, Bachet's Diophantus, Fundamental Theorem of Finite Calculus, Mean Value Theorem, Michael Psellus, Nicole Oresme New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 23
books:
See all 23 books this book cites
- Introduction to Arithmetic by G Nicomachus on page 26, page 42, and Back Matter
- The History of Philosophy (1701) by Thomas Stanley in Front Matter, and Back Matter
- Prime Time by Sandra Brown in Back Matter
- The Prime Numbers and Their Distribution (Student Mathematical Library, Vol. 6) (Student Mathematical Library, V. 6) by Gerald Tenenbaum in Back Matter
- Nicole Oresme and the Medieval Geometry of Qualities and Motions (Medieval Science Pubns., No. 12) by Marshall Clagett in Back Matter
See all 23 books this book cites
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