Is there a constant $c<1$ such that Riemann's $\zeta$ function is known not to vanish for $\Re s \geq c$ (i.e. real part at least $c$)?
My feeling is that the answer is "no", but I could not find a clear statement of this on the web or in books.
(any other enlightening basic information about the state of knowledge about zero free regions would be appreciated)