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

and , 2.5) pointed out that this implies that *S*(*T*)/(log log *T*)^{1/2} resembles a with mean 0 and variance 2π^{2}. In particular |*S*(*T*)| is usually somewhere around (log log *T*)^{1/2}, but occasionally much larger. The exact order of growth of *S*(*T*) is not known. There has been no unconditional improvement to Riemann's original bound *S*(*T*)=O(log *T*), though the Riemann hypothesis implies the slightly smaller bound *S*(*T*)=O(log *T*/log log *T*) (). The true order of magnitude may be somewhat less than this, as random functions with the same distribution as *S*(*T*) tend to have growth of order about log(*T*)^{1/2}. In the other direction it cannot be too small: showed that , and assuming the Riemann hypothesis Montgomery showed that .

## Examples of exactly solvable models are discussed in detail."A.

### One example of such a system is known as the bosonic Riemann gas.

Some of the arguments for (or against) the Riemann hypothesis are listed by , , and , and include the following reasons (most discussed in more detail in the rest of this article).

### Hutchinson, "Physics of the Riemann hypothesis", Rev.

The **Riemann hypothesis** is a mathematical . Many people think that finding a of the is one of the hardest and most important unsolved problems of .^{}

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This was included in Hilbert's famousList of Problems, and was partly resolved many years later:Cantor's Continuum Hypothesis is an "Undecidable Statement"of Set Theory.

## wrote a program to graph regions of the Riemann zeta function ..

In addition to developing theories, he also produced ingenious proofs;for example he was first to prove Miquel's n-Circle Theorem,and did so with a purely geometric argument.

## theory of Alternative Facts Donald J

Sylvester each invented it independentlya few years later.)Using insights from non-Euclidean geometry, he did importantmathematical work in a very wide range of physics;for example he improved Green's theorem, generalized the Laplace operator,studied gravitation in non-Euclidean space,and gave a new derivation of Maxwell's equations.

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(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 mini-bio,which Riemann extended via analytic continuation for all cases.