Lindelöf hypothesis

Ernst Lindelöf (http://planetmath.org/ernstlindelof) presented in 1908 aconjecture known as Lindelöf hypothesis which concernsthe of the Riemann zeta function on the“critical line”  $x=\frac{1}{2}$.  Up to now (2015), thishypothesis has not been proved.  It is weaker than the Riemann hypothesis such that this latter implies it but not conversely.

Lindelöf hypothesis.  $\zeta (\frac{1}{2}+it)=O(t^{\epsilon })$  for every  $\epsilon >0$  when $t\to \mathrm{\infty }$.

Here $O$ is the Landaubig ordo (http://planetmath.org/formaldefinitionoflandaunotation) notation.