Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire (PDF)

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The definitive story of the Riemann Hypothesis, a fascinating and epic mathematical mystery that continues to challege the world.In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world.

User’s Reviews

Editorial Reviews: Review “Derbyshire’s attempt to take nonmathematicians into this subject had me on the edge of my seat.”—Los Angeles Times “Riemann and his colleagues come to life as real characters and not just adjectives for conjectures and theorems.”—Scientific American About the Author JOHN DERBYSHIRE is a contributing editor for National Review, where he writes a regular column. He also contributes regularly to National Review Online and writes frequently for a number of other publications, including the Wall Street Journal, the American Conservative, the Washington Examiner, and the New Criterion. In addition to his opinion journalism, he writes on the subject of mathematics and is the author of the books Prime Obsession and Unknown Quantity. His novel, Seeing Calvin Coolidge in a Dream, was chosen as a New York Times Notable Book of the Year. A native of England, Derbyshire now lives on Long Island, New York, with his wife and two children.

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

⭐Bernhard Riemann was one of the greatest mathematicians of all time. Indeed, he is a strong candidate for GOAT. Mathematics, John Derbyshire tells us, is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. Riemann is most famous for his work in analysis. Analysis is the mathematics of the continuum. In addition Riemann made contributions to geometry — he was one of the discoverers of non-Euclidean geometry, which would eventually become the basis of Einstein’s Theory of General Relativity. And, most surprisingly, Riemann discovered a deep connection between analysis and arithmetic, number theory, the prime numbers. He set this out in one 1859 paper entitled “On the number of prime numbers below a given size”. Within that paper he made a guess, which he couldn’t prove, and wrote”One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation.”Over 160 years later that guess, now known as the Riemann Hypothesis, remains unproved. Derbyshire writes, accurately I think, “The Riemann Hypothesis is now the great white whale of mathematical research.”What is the Riemann Hypothesis? It is a very technical conjecture about a mathematical object known as the Riemann zeta function. I cannot explain it briefly. At least half of Prime Obsession is devoted to that purpose. You’re not going to get a comprehensible shorter answer than by reading this book. I thoroughly enjoyed it.Make no mistake — this is a book about math. If you don’t like math, you should probably not attempt it. However, it is also a book about mathematicians. The odd-numbered chapters cover the mathematics, while the even-numbered chapters cover history and biography. You could presumably read the even chapters alone to learn something about the Riemann Hypothesis while avoiding all the math. I didn’t try that, so I can’t tell you how well it would work.Derbyshire is careful to describe himself as a journalist, not a mathematician. However, he obviously knows a great deal about mathematics and is good at explaining it. I read The Music of the Primes at the same time as my first reading of Prime Obsession. The Music of the Primes is an attempt by Marcus du Sautoy, a card-carrying mathematician, to do what Derbyshire has done in Prime Obsession. I was surprised to find that Prime Obsession is much the better of the two books. It is not only that Derbyshire explains better — his explanations are also, surprisingly, more mathematically rigorous than du Sautoy’s.I was particularly impressed with Derbyshire’s handling of the theorem he grandly calls “The Golden Key”, also known as Euler’s product formula. He presents a complete and very clear proof. He explains how he found this proof as follows”When jotting down the ideas that make up this book, I first looked through some of the math texts on my shelves to find a proof of the Golden Key suitable for non-specialist readers. I settled on one that seemed to me acceptable and incorporated it. At a later stage of the book’s development, I thought I had better carry out authorial due diligence, so I went to a research library (in this case the excellent new Science, Industry and Business branch of the New York Public Library in midtown Manhattan) and pulled out the original paper from Euler’s collected works. His proof of the Golden Key covers ten lines and is far easier and more elegant than the one I had selected from my textbooks. I thereupon threw out my first choice of proof and replaced it with Euler’s. The proof in part III of this chapter is essentially Euler’s. It’s a professorial cliché, I know, but it’s true nonetheless: you can’t beat going to the original sources.”Derbyshire’s exposition of Euler’s proof covers far more than ten lines — Euler was writing for mathematicians and could abbreviate, knowing his readers would fill in the gaps. Derbyshire makes no such assumption and his proof is a thing of beauty.Mathematicians are concerned, more than any other profession I know (including the arts) with the pursuit of beauty. Nonmathematicians are often surprised to hear this — they don’t perceive beauty in mathematics. Derbyshire has done an outstanding job of presenting mathematics as the beautiful thing it is.

⭐”This isn’t magic. There’s a reason this stuff works,” my high school math teacher used to say. Of course, there are some contentions, hypotheses, in math where we don’t know if they work, if they are true.For professional mathematicians, one of the most important of these is the Riemann Hypothesis. Everlasting fame amongst mathematicians, and, incidentally, a million dollars is waiting for the person who can nail the truth of the “RH” down.Unlike some famous math problems, the gist of the RH is not readily apparent to most non-mathematicians. Derbyshire has to spend some time explaining what is meant by “All non-trivial zeros of the zeta function have real part one-half.” And, as someone whose formal math instruction ended with four years of high school math and who reads the very occasional popular math book by Gleick, Peterson, or Paulos, I’m pretty much the target audience Derbyshire pitches that explanation to.The book’s style reminded me of the science histories of James Burke. But where Burke’s work is a pinball version of history, caroming from person to person, theory to theory, Derbyshire’s is a train of mathematical explanation covering the work leading up to, and proceeding from, the RH. Occasionally, Derbyshire stops at some station, pulls up the blind, and looks at some area of tangential interest: famous mathematicians including Gauss, Hilbert, Russell, Dyson, and Turing (who thought RH untrue and attempted to build a computing device to disprove it); German educational reforms of the early 19th century; the Cambridge Five spies; and, most often, since this book is ostensibly a biography of him, the life of Bernhard Riemann. But it’s not long before we’re back on that math train again. This is not to shortchange the non-math interludes of the book. Derbyshire’s quick asides gave me a lot of ideas for further reading. And, if less than half of the book’s 422 pages cover Riemann’s life, you still get some idea of his protean mind so important not only to mathematics but modern physics.Derbyshire’s claim that, if you don’t understand the RH after he explains it you never will, seems credible. I won’t claim I immediately followed his chain of explanations the first time around. But that had more to do with trying to read this book in 15 minute intervals over a week rather than Derbyshire’s prose. Upon reviewing many sections again, things became clearer.The book briefly notes some of the consequences of RH, practical and theoretical. A lot of math is based on the assumption it’s true. And the RH may have some mysterious relation to the world of quantum physics. In the commercial and military worlds, where encryption methods based on prime numbers are important, the RH, which has to do with the distribution of primes, may have significant importance if proved true.I think one of the best things about this book is that, briefly, in a simple way, a non-mathematician like me can get some small idea of the excitement mathematicians feel upon discovering some curious pattern in the world of numbers.The only complaint I have with this book is its format. Is it too much to ask that, in the age of computerized typesetting and with an author whose footnotes are all worth reading, that we put those footnotes at the bottom of the relevant page and not at the end of the book?

⭐There are two excellent and very readable books that seek to demystify the Riemann Hypothesis, one is Marcus du Sautoy’s ‘The Music of the Primes’ and this is the other one. It starts off fairly gently and glides you through a lot of preliminary number theory stuff that you need to understand before you get to the Riemann Hypothesis itself. It helps if you have a good understanding of algebra, but even if you feel doubtful about that just know that John Derbyshire has a great way of explaining difficult concepts. The first half of the book is fairly easy to assimilate one chapter at a time, and even if this is all you read and understand you will have learned a lot about the prime number theorem, the zeta function, complex numbers and the Riemann Hypothesis; and it will also help you to understand the Euler Identity, an extremely beautiful equation. The latter part of the book has to explain some rather complex mathematics and it takes some time and concentration to get to grips with it. Don’t let that put you off. Just persevere by reading slowly and re-reading the chapters – going back to earlier chapters helps a lot. It is, after all, a complex mathematical concept. In the Prologue, John Derbyshire states that he aimed this book at the ‘intelligent and curious but non mathematical reader’. Anyway, you can also read this book as a historical account of the development of our understanding of prime numbers, a fascinating subject in itself.

⭐This is a very easy review to write. If you are looking at this review then you want to know something about the Riemann Hypothesis and how it relates to the Prime Numbers. Well this book takes you step-by-step from the very beginnings to the final conclusion in a very nicely written way. You will need a good “A”-level in Pure Mathematics (at least) to get all the maths bits, but just by reading the text you should gain a good understanding of the link between the RH and the Primes. Unless you are a degree level mathematician who can work their way through the original paper – then this book is probably your best route to understanding this topic.

⭐* PhysicalThe book i have has 422 pages of text, is printed on a fair size of font for those who require specs, and the paper is of reasonable quality too.* Target audience, A level / H.N.D, Undergraduate,Graduate, Post Graduate?This book is accessible when considering its quality of writing, but (i.m.h.o) this is better introductory read by second year mathematics students and above.* What’s the best topics then?With some books it is a historical journey with what they have done added. This book gives a fair explanation of the maths and his life.The more accessible bits of the Riemann contribution to mathematical knowledge is very well explained. The critical theory of arithmetic that all numbers greater than one are a compound of prime numbers is very well explored too. Zeta function, the one of the many function forms, the prime number generating function and its simpler proofs is as well explained as i have seen in other small number of samples. The way the calculation of the number of primes up to any integer is engaging to read. The many forms of the Zeta function, sums and products equivalence, ‘golden key’ the supporting fundamental math in this topic is really nicely completed. The writer knows his limits as this is a popular science, math book. And this needs to be limited to not overwhelm the general reader. But the math student in particular will find this book supportive of their topics and when, importantly, you access videos from a very well known video sharing website, (you know the one) you may find this knowledge comprehension greatly accelerated. By using both this book and the videos at whichever level of academic difficulty you’re at the moment, you may have a clearer time on this introductory level of the important math topic. Its encouraging me to have another look at further books too.* SummaryWith reading this book, ands watching and rewatching the videos on the very well known website that cover these same areas, you may find this a real interest and enjoyable way to gather knowledge in this area. This limited math level dexterity is fair, comprehensible, and really clearly written. You need the online videos, (and a reasonable bb connection), and (this) book together then you’re more likely to make progress. (i.m.h.o)

⭐I have read this book and two of the other three popularisations about the Riemann hypothesis. Instead of interviewing mathematicians who may be near to solving it or writing around the subject, this book actually works through the mathematics of Riemann’s 1859 paper. “Prime obsession” emphasises the centrality of the other parts of Riemann’s paper apart from the famous Hypothesis. By doing this it helps to explain why some 30 years later that mathematicians were able to prove the Prime Number Theorem, independently of the truth or otherwise of the famous Hypothesis. The Prime Number Theorem states, roughly that: as numbers get larger the number of primes less than that number tends to about the number divided by its logarithm (base e). The reason the Prime Number Theorem could be proved, irrespective of Riemann’s Hypothesis’ truth, is because of the techniques that Riemann invented in his 1859 paper.Riemann’s starting point was to generalise Euler’s formula which relates the sum of the reciprocals of natural numbers: 1+1/2+1/3+1/4+… to the product of the inverses of the prime numbers(1/2)*(1/3)*(1/5)*(1/7)*(1/11)*…..Derbyshire’s explanation is far clearer and much easier to follow than those in the other popularisations.This book is precise and clear: one really feels that one has some insight into an astonishing piece of creative mathematical work by the time one has read the book. That alone in my opinion should qualify it as one of the greatest pieces of popular science writing of this or any other decade.This book needs to be more actively marketed: whatever its faults, the author has made a genuine attempt to really explain a great piece of science technically to a non -technical audience, rather than just waffling around the subject and making us all feel these things are so far above our heads we will never understand them in any way. This courage on the author’s part needs to be more widely feted.I cannot do more than endorse the other reviewers’ praise for this classic-to-be. for those interested in pursuing this fascinating subject further, I found Gamma: Exploring Euler’s Constant (Princeton Science Library) by Havil to be a wonderful book.

⭐This is easily the best introduction to the subject I have seen. Each chapter is followed by one on the history which makes it accessible to everyman. This history is fascinating, a thing which can be done well or badly: it is done very well! All the main equations are within the grasp of the enthusiast with no more knowledge than A level. The graph of the zeta function is fundamental and it is very well done here.What gives the book its intense narrative drive is the nature of the puzzle: that the non-trivial zeros are all on the line with real part 1/2, along with the fact that it is very clearly explained. That the Riemann Hypothesis gives every sign of being on the point of solution after so many advances simply enhances the interest. It is not impossible that the final answer will be achieved by a reader of this book who does not have the mindset of everyone else who has been engaged with it for years.I believe that some areas of this book could be more accessible yet. The graph and the plotting of points is still difficult. Maybe some software should be provided which shows other dimensions than the two in a book. This author has done a magnificent job and may yet do it better, or someone else. Once it falls within the range of far more minds one of them might just have the originality to make a new move and fresh insights which explain this at last.I was enthralled by this book. Compared with other efforts, this one is a phenomenal achievement.William Scott

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