Can anyone give me a reference for the following theorem on the Riemann zeta function?

If the Lindelof Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then there are only finitely many zeros of $\zeta(s)$ off the critical line $Re(s)=\frac{1}{2}$.

I've heard that this is the case (and I think I read it on the internet, but can't find it again), but I'm looking for an actual proof.

Many thanks for any help with this!