holds for all if and only if the Riemann hypothesis holds.

A precise version of Koch's result, due to , says that the Riemann hypothesis is equivalent to

Talk:Riemann hypothesis - Wikipedia

Unfortunately, it looks like in this case it’s not a real proof of the Riemann Hypothesis, and this post on a Nigerian discussion forum says emphatically. As mentioned in that post, there’s under his name, which is actually a copy of a paper by someone called Werner Raab (retired). Raab’s website – is empty, and has a single broken link to “the truth of the Riemann hypothesis”. Some digging reveals that . Confusingly, Enoch seems to be.

proved that the Riemann Hypothesis is true if and only if the space of functions of the form

Riemann Hypothesis - Wikipedia | Conjecture | Analysis

for every positive ε is equivalent to the Riemann hypothesis (). (For the meaning of these symbols, see .) The determinant of the order n is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. The Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger

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

Riemann Hypothesis - Art of Problem Solving

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.

Riemann hypothesis - The Full Wiki

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 .

Grand Riemann hypothesis | Wiki | Everipedia

constructed a natural space of invariant functions on the upper half plane which has eigenvalues under the Laplacian operator corresponding to zeros of the Riemann zeta function, and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space the Riemann hypothesis would follow. discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator.

Riemann Hypothesis | Brilliant Math & Science Wiki

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 .

Riemann Hypothesis - Wikibooks, open books for an …

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 p-adic L-function with the eigenvalues of an operator, so can be thought of as an analogue of the Hilbert–Pólya conjecture for ().