Functions: Zeta

"No one has ever been able to prove, for example, that every even number greater than two can be expressed as the sum of two primes. Yet this is as well established by observation as any of the laws of physics. It is known that this and various other theorems are true if a certain hypothesis about the Zeta function, enunciated by Riemann nearly a century ago, is correct. No one has been able to prove this hypothesis. It has only been shown that all the consequences deducible if it is true are so far verified by experience. But any day a computer with little knowledge of pure mathematics may disprove it. Here then is a possible triumph for the mathematical amateur." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"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 zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. [...] The main interest comes from trying to improve the Prime Number Theorem, i.e., getting better estimates for the distribution of the prime numbers. The secret to the success is assumed to lie in proving a conjecture which Riemann stated in 1859 without much fare, and whose proof has since then become the single most desirable achievement for a mathematician." (Martin C Gutzwiller, "Chaos in Classical and Quantum Mechanics", 1990)

"I am firmly convinced that the most important unsolved problem in mathematics today is the truth or falsity of a conjecture about the zeros of the zeta function, which was fi rst made by Riemann himself. [...] Even a single exception to Riemann’s conjecture would have enormous ly strange consequences for the distribution of prime numbers [...] If the Riemann hypothesis turns out to be false, there will be huge oscillations in the distribution of primes. In an orchestra, that would be like one loud instrument that drowns out the others - an aesthetically distasteful situation." (Enrico Bombieri, "Prime Territory: Exploring the Infinite Landscape at the Base of the Number System", The Sciences, 1992)

"It is intriguing that any of the various new expansions and associated observations relevant to the critical zeros arise from the fi eld of quantum theory, feeding back, as it were, into the study of the Riemann zeta function. But the feedback of which we speak can move in the other direction, as techniques attendant on the Riemann zeta function apply to quantum studies." (Jonathan M Borwein et al, "Computational Strategies for the Riemann Zeta Function", Journal of Computational and Applied Mathematics Vol. 121, 2000

"Riemann had found a passageway from the familiar world of numbers into a mathematics which would have seemed utterly alien to the Greeks who had studied prime numbers two thousand years before. He had innocently mixed imaginary numbers with his zeta function and discovered, like some mathematical alchemist, the mathematical treasure emerging from this admixture of elements that generations had been searching for. He had crammed his ideas into a ten-page paper, but was fully aware that his ideas would open up radically new vistas on the primes. (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"[...] in one of those unexpected connections that make theoretical physics so delightful, the quantum chorology of spectra turns out to be deeply connected to the arithmetic of prime numbers, through the celebrated zeros of the  Riemann zeta function: the zeros mimic quantum energy levels of a classically chaotic system. The connection is not only deep but also tantalizing, since its basis is still obscure - though it has been fruitful for both mathematics and physics." (Michael V. Berry)

"How could it be that the Riemann zeta function so convincingly mimics a quantum system without being one?" (Michael V Berry) [33] 

"It’s a whole beautiful subject and the Riemann zeta function is just the fi rst one of these, but it’s just the tip of the iceberg. They are just the most amazing objects, these L-functions - the fact that they exist, and have these incredible properties are tied up with all these arithmetical things - and it’s just a beautiful subject. Discovering these things is like discovering a gemstone or something. You’re amazed that this thing exists, has these properties and can do this." (J Brian Conrey)