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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.
The Riemann hypothesis was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler ), behind which a theta function lies.
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].
Riemann's statement of the Riemann hypothesis, from his 1859 paper. [11] (He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line. He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line.
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.
The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1/2. The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1 / 2 .
the Riemann hypothesis is equivalent to the statement that > for every positive integer . The numbers (sometimes defined with a slightly different normalization) are called Keiper-Li coefficients or Li coefficients. They may also be expressed in terms of the non-trivial zeros of the Riemann zeta function: