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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.
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.
It asks for more work on the distribution of primes and generalizations of Riemann hypothesis to other rings where prime ideals take the place of primes. Absolute value of the ζ-function. Hilbert's eighth problem includes the Riemann hypothesis, which states that this function can only have non-trivial zeroes along the line x = 1/2 [2].
The Riemann ξ function is given by = / ()where ζ is the Riemann zeta function.Consider the sequence = ()! [ ()] | =. Li's criterion is then the statement that the Riemann hypothesis is equivalent to the statement that > for every positive integer .
Development of the explicit formulae for a wide class of L-functions was given by Weil (1952), who first extended the idea to local zeta-functions, and formulated a version of a generalized Riemann hypothesis in this setting, as a positivity statement for a generalized function on a topological group.
Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968).
In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the weak Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes.
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 ...