holds for all if and only if the Riemann hypothesis holds.
The greatest problem for mathematicians now is probably the Riemann Hypothesis. Andrew Wiles
Riemann Hypothesis  Clay Mathematics Institute
Another example was found by showing that the Riemann hypothesis is equivalent to a statement that the terms of the are fairly regular. More precisely, if F_{n} is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0
The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n). A typical example is (), which states that if σ(n) is the , given by
A proof of the Riemann Hypothesis?! • r/math  reddit
The Riemann hypothesis implies that the zeros of the zeta function form a , meaning a distribution with discrete support whose Fourier transform also has discrete support. suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1dimensional quasicrystals.
Louis de Branges () showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions. However showed that the necessary positivity conditions are not satisfied.
A proof of the Riemann Hypothesis?
is actually an instance of the Riemann hypothesis in the function field setting. This led to conjecture a similar statement for all ; the resulting were proven by (, ).
introduced global zetafunctions of (quadratic) and conjectured an analogue of the Riemann hypothesis for them, which has been proven by Hasse in the genus 1 case and by in general. For instance, the fact that the , of the quadratic character of a of size q (with q odd), has absolute value
but proving the Riemann Hypothesis in 10 pages

the proof of the Riemann Hypothesis will follow the same strategy.
proved that the Riemann hypothesis is equivalent to the "best possible" bound for the error of the .

The Riemann Hypothesis  Karl Sabbagh  Macmillan
is equivalent to the Riemann hypothesis. Here is the number of terms in the Farey sequence of order n.

The Riemann Hypothesis The Greatest Unsolved Problem in Mathematics
Andrew Wiles Quote: The greatest problem for mathematicians now is probably the Riemann Hypothesis.
Quasicrystals and the Riemann Hypothesis  …
Hilbert and Polya suggested that one way to derive the Riemann hypothesis would be to find a , from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies the criterion on real . Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a on an group, the zeros of a are eigenvalues of a of a Riemann surface, and the zeros of a correspond to eigenvectors of a Galois action on .
Quasicrystals and the Riemann Hypothesis
Several mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been accepted as correct solutions. lists some incorrect solutions, and more are .
Claimed Proof of Riemann Hypothesis  Slashdot
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 ().
Claimed Proof of Riemann Hypothesis
The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first cohomology group of the Spec(Z) of the integers. described some of the attempts to find such a cohomology theory.
Proving the Riemann Hypothesis and Impact on Cryptography
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.