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The generalized Riemann hypothesis
(Asked what he would first do, if he were magically awakened aftercenturies, David Hilbert replied "I would ask whetheranyone had proved the Riemann Hypothesis.")ζ(.) was defined for convergent cases in Euler's minibio,which Riemann extended via analytic continuation for all cases.
On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.
Lagarias Complements of Li`s criterion for the Riemann Hypothesis
The extends the Riemann hypothesis to all of . 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 Lfunctions of of number fields.
Berry has reformulated the RiemannHypothesis in terms of a search for a dynamical system with a veryparticular set of properties.
A.
holds for all if and only if the Riemann hypothesis holds.
There are of zeta functions with analogues of the Riemann hypothesis, some of which have been proved. of function fields have a Riemann hypothesis, proved by . The main conjecture of , proved by and for , and Wiles for , identifies the zeros of a padic Lfunction with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for ().
Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no nontrivial integer linear relations between their imaginary parts). of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorize as a product of powers of , so zeros of Artin Lfunctions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the Lfunctions of some : these can have multiple zeros at the real point of their critical line; the predicts that the multiplicity of this zero is the rank of the elliptic curve.
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some reformulations of the Riemann Hypothesis
We formulate the Riemann hypothesis physically as a nonzero condition on the transition amplitude between two special states associated with the point of origin and a point half way around the cylinder each of which are fixed points of a $Z_2$ transformation. By studying partial sums we show that that the transition amplitude formulation is analogous to neutrino mixing in a low dimensional context.
Has Riemann Hypothesis been solved ?  Yahoo Answers
al. for the Riemann Hypothesis revisited using similar functions" (preprint 01/06)[abstract:] "The original criteria of Riesz and of HardyLittlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of BaezDuarte concerning a representation of the Riemann Zeta function.
On Nicolas criterion for the Riemann Hypothesis  …
Merlini, (preprint 04/04)[abstract:] "We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov.
when Hilbert (transform) meets Riemann (hypothesis ..
where Hardy's function and the θ are uniquely defined by this and the condition that they are smooth real functions with θ(0)=0. By finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line. To verify the Riemann hypothesis up to a given imaginary part T of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region and checking that it is the same as the number of zeros found on the line. This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).
The Riemann Hypothesis, explained – Jørgen Veisdal  …
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 or , imply that they expect (or at least hope) that it is true. The few authors who express serious doubt about it include who lists some reasons for being skeptical, and who flatly states that he believes it to be false, and that there is no evidence whatever for it and no imaginable reason for it to be true. The consensus of the survey articles (, , and ) 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.
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