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



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> If the Lindelöf 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 $Re(s)=\frac{1}{2}$.
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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.



Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?
The Lindelof hypothesis (LH) does not seem to give such precise information about the zeros of $\zeta(s)$. Here are three known implications of Lindelof on the zeros, but they will be seen to fall far short of showing finitely many exceptions to RH. Thus, no result of the form stated in the question exists in the literature.


1. Backlund showed that LH implies that for large $T$ and any $\epsilon >0$ there are at most $o(\log T)$ zeros of $\zeta(s)$ with real part bigger than $1/2+\epsilon$ and imaginary part between $T$ and $T+1$. (For comparison there are about constant times $\log T$ zeros of $\zeta(s)$ with imaginary part between $T$ and $T+1$.)
2. LH implies the density hypothesis: For any $\sigma>1/2$, the number of zeros of $\zeta(s)$ with real part $\ge \sigma$ and imaginary part between $0$ and $T$ is denoted by $N(\sigma,T)$. Then LH implies the bound $N(\sigma,T) = O(T^{2(1-\sigma)+\epsilon})$.
3. A theorem of Halász and Turán: LH implies that $N(3/4+\epsilon,T)= O(T^{\epsilon})$.


These results may be found in the books of Titchmarsh, Ivic or Edwards on zeta.


