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That is the Riemann Hypothesis.
is valid for every s with real part greater than ½, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the is defined by
The Clay Mathematics Institute offers $1,000,000 for a proof of the Riemann Hypothesisan extremely thorough mathematical descriptionof the Riemann Hypothesis (with historical background, etc.) provided by Enrico Bombieri for the purposes of this competitionvideorecording of an introductory lecture by J.
for all n 5040 if and only if the Riemann hypothesis is true.
A major breakthrough came in 1896, when two mathematicians independently proved the Prime Number theorem, which showed not just that prime numbers become rarer as they grow larger, but that their reoccurrence follows a specific formula (for those who care, the average gap between consecutive prime numbers is roughly proportional to the logarithm). The Prime Number theorem also encouraged mathematicians to begin working on Riemann's hypothesis, which gives a far more detailed picture of how the primes are distributed and which has consequences, not just for prime numbers, but also for many other areas of mathematics, including the analytic properties of functions and representations of nonsingular cubic forms.
The fame of the Riemann hypothesis has grown steadily over the past 150 years. In 1900, David Hilbert, a German mathematician and contemporary of Einstein's, in a famous speech to the International Congress of Mathematicians, presented a list of what he considered to be the 23 most important problems for the new century. Hilbert's problems came from a variety of mathematical fields, and were to have a significant influence on 20thcentury mathematics. One hundred years later, the Clay Mathematics Institute, which was founded by Landon T. Clay, a Boston businessman, published its list of what it called the seven millenniumprize problems, and promised a $1m reward for solving any one of them. The Riemann hypothesis appears on both lists.
holds for all if and only if the Riemann hypothesis holds.
Riemann's riddle has become a fixture on the mathematical landscape. Over the years, hundreds of results have been published that assume the truth of the hypothesis, but a proof of the conjecture would have immense consequences. Equally, a disproof would be the mathematical equivalent of an earthquake, destroying decades of work at a stroke.
Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in solving it. Some typical examples are as follows.
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Spiliopoulos', Introduction to the Riemann HypothesisG.
All three of these books offer fascinating accounts of the story surrounding the Riemann hypothesis. Karl Sabbagh approaches the subject in the manner of an anthropologist, recording dozens of conversations with mathematicians working on various aspects of the problem. As a nonmathematician, he appreciates the difficulty of explaining a recondite mathematical problem to a general audience, and gives a vivid account both of the passion that mathematicians have for their subject and of the fascination for this particular problem.
Perry's introductory notes on the Riemann HypothesisP.
The prime number theorem implies that on average, the between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case when even the best bound that can currently be proved using the Riemann hypothesis is far weaker than what seems to be true: implies that every gap is O(log(p)^{2}) which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture ().
Hutchinson, "Physics of the Riemann hypothesis", Rev.
is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is , a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.
Riemann Hypothesis  Clay Mathematics Institute
The most detailed account is given by Marcus du Sautoy. A professor of mathematics at Oxford University, Mr du Sautoy provides an engaging and accessible history of work on prime numbers and the Riemann hypothesis. He also has an eye for modern applications, and offers detailed discussion of the relevance of the Riemann hypothesis to cryptographic security as well as an interesting account of its possible links with quantum physics.
The Riemann Hypothesis, explained – Jørgen Veisdal  …
The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that
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