Artin L-functions (2005)(en)(1s)
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Godement and H.
This Week's Finds in Mathematical Physics (Week 216)
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Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta-function connects through its values at positive even integers and negative odd integers to the Bernoulli numbers , one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for p -adic L -functions , which describe certain Galois modules.
The statistics of the zero distributions are of interest because of their connection to problems like the Generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest. The fractal structure of the distributions has been studied using rescaled range analysis. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function , and also for the zeros of other L-functions of different orders and conductors.
One of the influential examples, both for the history of the more general L -functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the s. It applies to an elliptic curve E , and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers or another global field : i.
L-function - Wikipedia
Much previous work in the area began to be unified around a better knowledge of L -functions. This was something like a paradigm example of the nascent theory of L -functions. This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin L -functions , which, like Hecke L -functions , were defined several decades earlier, and to L -functions attached to general automorphic representations.
Gradually it became clearer in what sense the construction of Hasse—Weil zeta-functions might be made to work to provide valid L -functions, in the analytic sense: there should be some input from analysis, which meant automorphic analysis. Some calculations of zeros of the zeta function are listed below. So far all zeros that have been checked are on the critical line and are simple. A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros. Gram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation Gram's law.
The first failure of Gram's law occurs at the 'th zero and the Gram point g , which are in the "wrong" order. The indices of the "bad" Gram points where Z has the "wrong" sign are , , , , A Gram block is an interval bounded by two good Gram points such that all the Gram points between them are bad. For example, the interval bounded by g and g is a Gram block containing a unique bad Gram point g , and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions.
This means that both rules hold most of the time for small T but eventually break down often. Indeed, Trudgian showed that both Gram's law and Rosser's rule fail in a positive proportion of cases. Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authors who express an opinion, most of them, such as Riemann or Bombieri , imply that they expect or at least hope that it is true. The consensus of the survey articles Bombieri , Conrey , and Sarnak is that the evidence for it is strong but not overwhelming, so that while it is probably true there is some reasonable doubt about it.
From Wikipedia, the free encyclopedia. Conjecture in mathematics linked to the distribution of prime numbers. For the musical term, see Riemannian theory. He was discussing a version of the zeta function, modified so that its roots zeros are real rather than on the critical line. 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.
Burton , p. Main article: Selberg zeta function. Main article: Selberg's zeta function conjecture. Variae observationes circa series infinitas.
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