Riemann Hypothesis

Hypothesis

All the nontrivial zeroes of the analytic continuation of the Riemann zeta function have a real part equal to $\frac 1 2$.

Trivial zeroes occur at every negative even integer ($-2, -4, -6$ etc.).

Critical Line

The line defined by the equation $\displaystyle z = \frac 1 2 + i y$ is known as the critical line.

Hence the popular form of the statement of the Riemann hypothesis:

"All the nontrivial zeroes of the Riemann zeta function lie on the critical line."

Comment

This problem is the first part of no. 8 in the Hilbert 23, and also one of the Millennium Problems, the only one to be in both lists.

By using the WKB theory of quantum mechanics one can prove that

$\displaystyle \frac{ \xi(s)}{\xi(0)}= \frac{det(H+s(s-1)+1/4)}{det(H+1/4)}$

with the potential inside the Hamiltonian given by:

$V^{-1}(x)= 2 \sqrt \pi \dfrac{d^{1/2}N(x)}{dx^{1/2}}$

with

$\pi N(x) = \operatorname{Arg} \xi(1/2+i \sqrt x)$

to get the potential we simply compare the 'spectral' function \( \int_{0}^{\infty}exp(-sx)dN(x) \) and its classical counterpart \( \frac{1}{\sqrt 2\pi s}\int_{0}^{\infty}dxexp(-sx)dV(x) \) taking the inverse Laplace transform on both sides we get the inverse of the potential as a function of the Eigenvalue Staircase \( N(x) \)

Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.