Prime Numbers and the Riemann Hypothesis

by Barry Mazur

Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann Hypothesis, which remains to be one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann Hypothesis. Students with minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann Hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann Hypothesis.

Genres: Mathematics, Science, Nonfiction, Number Theory

156 pages, Paperback

First published October 18, 2015

About the author

Barry Mazur (Harvard University)

Reviews

[Esraa Gibreen · Rating 4 out of 5]

الكتاب مبسط لدرجة كبيرة ويصلح كمقدمة عن فرضية ريمان -أحد المسائل المفتوحة التي لم تحل إلى الآن- وعلاقتها بالأعداد الأولية. أسلوبه بعيد كل البعد عن أسلوب الكتب الأكاديمية، حتى إن الكاتب يجتهد في بعض المواضع ليكون مضحكا. مقسم لعدد كبير من الفصول على صغر حجمه الذي لا يتعدى 150 صفحة، لذلك يوجد عدد لا بأس به من الفصول المكونة من صفحتين لا أكثر، وهذا تأكيدا لنهجه المثير لمتابعة قراءته وبعده عن جفاء الكتب الأكاديمية.

[Mark Moon · Rating 4 out of 5]

Cute intro to the prime-counting function and how it is related to the Riemann zeta function. I learned some new stuff but wasn't overly taxed; I'd classify it as "hard pop-sci". Lots of good pictures/graphs.

[Chayan · Rating 4 out of 5]

An amazing intro to Riemann Hypothesis and zeta functions. This 4 parts book starts with anecdotes and curious behaviour of numbers, especially primes, and moves on to zeta functions and the RH, taking a tour through generalised functions and Fourier transformation. It's a short but well researched book, not only in content, but in organisation, so that it feels almost effortless to move through the chapters. Despite the highly technical nature, the approach is seldom clinical. It builds suspense, piques interest and not unlike a thriller, leaves the reader to ponder.

[RoWoSthlm · Rating 4 out of 5]

A kind of rendezvous with the lady math and probably her most loved children - prime numbers, where I realized this stuff still attracts me, but some of it was really hard to grasp. I found some beautiful pieces in the book and got a better idea of how useful Riemann could actually one day become, when someone hugely smart gets it. As author points out, we been dealing with numbers for over a couple of millennia and yet we are in infancy of our understanding. Lucky are those who try.

[Fernando Pestana da Costa · Rating 4 out of 5]

This book is an excellent and gentle introduction to one of the most important unsolved problems in Mathematics: the Riemann Hypothesis. Written at a level that can be read by anyone with a moderately low sophisticated knowledge of mathematics (not more than the level of a first year undergraduate), accompanied by many mathematical asides and historical observations, and illustrated by a huge number of pictures and graphs that make for an important visual aid to the concepts being discussed, the book is a wonderful guide for everyone wanting to travel the pathway between the basic notion of a prime number, and some of the most difficult problems and sophisticated concepts in mathematics, and all this with the minimum possible technical stuff.

[Jeffrey · Rating 3 out of 5]

Of you are knowledgeable enough to have hear of the Riemann Hypothesis, then this is a great book for you. Part one explains what the Riemann Hypothesis is and why it is important to anyone who uses prime numbers. Part two goes into details of what the proof would be contingent on. Part three are tools and lemmas you would need to understand to complete the proof. Part three was over my head. So don't look for me to solve Riemann's hypothesis.
Book is technically proficient and challenging. A bit dry in the second and third parts. Could have used more documented anecdotes about who Riemann was and who has come close to proving his hypothesis.

[Joseph Schrock · Rating 5 out of 5]

“Prime Numbers and the Riemann Hypothesis” was a highly informative book for me. The last couple of parts of the book were quite a challenge for me to wade through, but I read all the way to the end.

I confess to not fully understanding nearly everything I read, but then, I’m a mere mathematical hobbyist – not a professional mathematician. Even though I muddled through four semesters of calculus (some 20 years ago), I found this book to be a genuine challenge – especially in the latter sections.

The book is divided in such a manner that those who’ve had no calculus can read the early parts of the book and get a rough idea of the essence of Riemann’s famous hypothesis. If one has an interest in Riemann’s work, this book is a good place for a non-professional reader to get a taste of the famous work.

[Diogo Silva · Rating 4 out of 5]

First half of the book is accessible and beautiful if you are interested in the topic.
Parts 2 and 3 were difficult to follow for me - the author warns about this but I feel a bigger effort of step by step explanation could have been made.
The conclusion part is again beautiful if you enjoy the topic.

[Patrick Underdown · Rating 2 out of 5]

A little too dense for me.

[Dave · Rating 5 out of 5]

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Good introduction for interested readers of all math backgrounds. His use of fourier transforms and explaining how Riemann had used them.

[Michiel · Rating 3 out of 5]

A layman's introduction to the Riemann hypothesis and how it relates to the distribution of prime numbers. Is more a dry summary of the main ideas than popular science in nice prose.

[Kent Sibilev · Rating 4 out of 5]

It is a very gentle introduction to Riemann Hypothesis. If you like this subject, John Derbyshire’s Prime Obsession book is much better read in my opinion.

[Saju Pillai · Rating 4 out of 5]

More a long paper than a book, but very well written.