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The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms.
Riesz showed that the Riemann hypothesis is equivalent to the claim that the above is true for any e larger than /. [1] In the same paper, he added a slightly pessimistic note too: « Je ne sais pas encore decider si cette condition facilitera la vérification de l'hypothèse » ("I can't decide if this condition will facilitate the ...
Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1 / 2 . In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1 / 2 + yi where y is a real number.
The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
More recent work by Alain Connes has gone much further into the functional-analytic background, providing a trace formula the validity of which is equivalent to such a generalized Riemann hypothesis. A slightly different point of view was given by Meyer (2005), who derived the explicit formula of Weil via harmonic analysis on adelic spaces.
The extended Riemann hypothesis asserts that for every number field K and every complex number s with ζ K (s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2. The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.
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. [2]
Specific choices of give different types of Riemann sums: . If = for all i, the method is the left rule [2] [3] and gives a left Riemann sum.; If = for all i, the method is the right rule [2] [3] and gives a right Riemann sum.