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Riemann Zeta Function
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In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in number theory because of its relation to the distribution of prime numbers. It ... has applications in other areas such as physics, probability theory, and applied statistics.
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The Riemann Zeta function creates a three dimensional graph. At the zeros of this graph are information directly tied to primes (Mathworld has a VERY in-depth explanation. Definitely not a casual read). The formula’s most common form is (for the real part only):
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The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta-function universality states that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.
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The book is aimed at providing a source of reference material for researchers requiring results associated with the Zeta function and related series. Detailed and methodical presentation of techniques for handling zeta type series is given and placed in a historical context giving the reader an appreciation of its origins. An extensive list of references is provided together with tables of identities demonstrating the wealth and depth of many results of series and integrals satisfied by Zeta related functions.
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It is based on statistical mechanics and the construction of a quantum statistical system whose partition function is the Riemann zeta function. This space X already appears in a very implicit manner in the work of Tate and Iwasawa on the functional equation. It is a noncommutative space in that, even at the level of measure theory, it is a tricky quotient space. For instance at the measure theory level, the corresponding von Neumann algebra, (3) R01 = Linfinity(A) X| k* where A is endowed with its Haar measure as an additive group, is the hyperfinite factor of type IIinfinity ...[according to Week 175 by John Baez, "... there is only one that is hyperfinite. You can get this by tensoring the type Iinfinity factor and the hyperfinite II1 factor.
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Computation of two billion zeros of the Zeta function at each of the height 1013, 1014, ΒΌ, 1023 and 1024 were done. What follows is a synthetic view of what is contained in the paperpaper, where the reader should refer for more details.
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Riemann Zeta Function