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

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 $Re(s)=\frac{1}{2}$.

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.

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    $\begingroup$ No such result exists in the literature! One beautiful result of Halasz and Turan says that LH implies that there are at most $T^{\epsilon}$ zeros of zeta with height up to $T$ and real part $>3/4$. This may give some indication of how far we are from LH implying all but finitely many zeros on the line. $\endgroup$
    – Lucia
    Jan 24, 2014 at 16:35
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    $\begingroup$ @Lucia: Please post your comment as an answer, so that this question can be closed. $\endgroup$
    – GH from MO
    Jan 24, 2014 at 20:47

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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.

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  • $\begingroup$ Thanks for this. I've looked in the books you mention, but I was hoping there was some more recent result! If nobody posts to say the result I mentioned has been proved, I'll accept your answer. $\endgroup$
    – Harry Macpherson
    Jan 28, 2014 at 22:18
  • $\begingroup$ @Lucia: How does one prove that LH implies the density hypothesis? I've checked the three references and I can't find this implication there. Also, I can see that the Bohr Landau theorem is essentially an averaged version of LH, in the sense that it implies LH "on average" if one takes $\epsilon$ in Backlund's version to be $1/\log T$. Thank you. $\endgroup$
    – Kevin Smith
    Nov 8, 2017 at 22:23
  • $\begingroup$ @Lucia - I mean the $\epsilon$ implies by the $o(\log T)$ statement, not the closeness to the line $\sigma=1/2$ $\endgroup$
    – Kevin Smith
    Nov 8, 2017 at 22:28
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    $\begingroup$ I don't know which statement you want. For parts (2) and (3), see Chapter 11 of Ivic's book (and the end of chapter notes). For part (1), see Chapter XIII of Titchmarsh. $\endgroup$
    – Lucia
    Nov 8, 2017 at 22:39

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