On Hypotheses: The Riemann Hypothesis

"Whoever proves or disproves [the Riemann Hypothesis] will cover himself in glory..." (Eric T Bell, 1937)

"[...] the Riemann hypothesis remains one of the outstanding challenges of mathematics, a prize which has tantalized and eluded some of the most brilliant mathematicians of this century...Hilbert is reputed to have said that the first comment he would make after waking at the end of a thousand year sleep would be, 'Is the Riemann hypothesis established yet?'" (Richard E  Bellman, A Brief Introduction of Theta Functions, 1961)

"At this point, it is not possible to remain silent on what is probably the most intriguing unsolved problem in the theory of the zeta function and actually in all of number theory - and most likely even one of the most important unsolved problems in contemporary mathematics, namely the famous Riemann hypothesis. [...] Still, the problem is open and fascinates and teases the best contemporary minds." (Emil Grosswald, "Topics in the Theory of Numbers", 1966)

"The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers. This fact alone singles out the Riemann hypothesis as the main open question of prime number theory." (Enrico Bombieri,  "Prime Territory", The Sciences,  1992)

"The Riemann hypothesis [...] is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory." (Bruce Schechter, "143-year-old problem still has mathematicians guessing", 2002)

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated." (Marcus du Sautoy, "The Music of the Primes", 2003)

"Solving any of the great unsolved problems in mathematics is akin to the first ascent of Everest. It is a formidable achievement, but after the conquest there is sometimes nowhere to go but down. Some of the great problems have proven to be isolated mountain peaks, disconnected from their neighbors. The Riemann hypothesis is quite different in this regard. There is a large body of mathematical speculation that becomes fact if the Riemann hypothesis is solved. We know many statements of the form “if the Riemann hypothesis, then the following interesting mathematical statement”, and this is rather different from the solution of problems such as the Fermat problem."  (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007) 

"Why is the Riemann hypothesis so important? Why is it the problem that many mathematicians would sell their souls to solve? There are a number of great old unsolved problems in mathematics, but none of them has quite the stature of the Riemann hypothesis. This stature can be attributed to a variety of causes ranging from mathematical to cultural. As with the other old great unsolved problems, the Riemann hypothesis is clearly very difficult. It has resisted solution for 150 years and has been attempted by many of the greatest minds in mathematics." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007) 

"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes." (David M Burton, "Elementary Number Theory", 2006)

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by numbers: music and maths", 2011)

"If [the Riemann Hypothesis is] not true, then the world is a very different place. The whole structure of integers and prime numbers would be very different to what we could imagine. In a way, it would be more interesting if it were false, but it would be a disaster because we've built so much round assuming its truth." (P  Sarnak)

"The Riemann Hypothesis is a precise statement, and in one sense what it means is clear, but what it's connected with, what it implies, where it comes from, can be very unobvious." (M Huxley)

"[...] the Riemann Hypothesis will be settled without any fundamental changes in our mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea is missing." (Y Motohashi)

"The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these elementary objects of arithmetic. And to have tools to study the distribution of these of objects." (H Iwaniec)