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The Lindelöf hypothesis is: $$ \forall \epsilon >0,\exists C_\epsilon >0,\forall t\ge 1,\quad \vert\zeta(\frac12+it)\vert\le C_\epsilon t^\epsilon.\qquad \tag{LH}. $$ It is a weaker statement than the Riemann hypothesis: $(RH)\Longrightarrow (LH)$. The (not-so-easy) texbook result that the estimate above is true for $\epsilon=1/6$ was improved in 1986 by Bombieri and Iwaniec (mathreview#:MR0881101) who found the estimate for $\epsilon=\frac{9}{56}$. Several works followed, using some variations of their method, but I do not think that the threshold $1/7$ was reached.

Now my question: is there a stronger inequality, e.g. $$ \exists C >0,\forall t\ge 2,\quad\vert\zeta(\frac12+it)\vert\le C(\ln t)^C\tag{LH$^\sharp$} $$ which would be equivalent to $(RH)$?

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  • $\begingroup$ Thanks for the answers. Let me reformulate my question: the estimate $$ \ln\bigl(\vert\zeta(\frac{1}{2}+it)\vert\bigr)=O\bigl((\ln t\ln\ln t)^{1/2}\bigr) $$ seems compatible with the remarks, but I guess that its logical relationship with $(RH)$ is not clear. $\endgroup$
    – Bazin
    Commented Sep 6, 2012 at 17:21
  • $\begingroup$ Something that I like with $(LH)$: you can try to improve the $\epsilon$, whereas $(RH)$ seems a different sort of game, all or nothing, no gradual approach. $\endgroup$
    – Bazin
    Commented Sep 6, 2012 at 17:23
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    $\begingroup$ Regarding your 2nd com.: For RH one could prove ever larger zero free regions, or ever larger proportions of zeros on the line. Regarding 1st: I am a quite a bit out of my comfort-zone commenting on this, so I only comment and do/did not include it in answer. I think (but this could be wrong) that there might not be such a link; the reason being that while zeros of zeta are of course important for the behavior on the critical line they are more so in a commulative sense. Thus it might be that if RH would only fail by very little this might not have enough influence to detect it this way. $\endgroup$
    – user9072
    Commented Sep 6, 2012 at 18:17
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    $\begingroup$ Just a historical remark: there were many exponents (at least 10) between Weyl's 1/6 and Bombieri-Iwaniec's 9/56, see p.118 in Titchmarsh: The theory of the Riemann zeta function (2nd edition, Oxford, 1986). $\endgroup$
    – GH from MO
    Commented Sep 7, 2012 at 2:17

3 Answers 3

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No. In fact it is known that for any $\varepsilon>0$ there are arbitrarily large values of $t$ with $$ |\zeta(1/2+it)| \ge \exp\left( (1-\varepsilon) \sqrt{\frac{\log t}{\log \log t}} \ \right).$$ This is a recent result of Soundararajan, Math. Ann. (2008) 342:467–486.

Ramachandra and Balasubramanian had previous proved an inequality of a similar form with the factor of $1-\varepsilon$ replaced by a smaller constant.

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No, zeta will not be so small, as already mentioned in another answer. Since you give the equation just as example, I assume you care for what is believed to be 'the truth'; thus in addition I mention a rather recent conjecture on the 'exact' maximal value (Farmer, Gonek, Hughes, The maximum size of L-functions, J reine angew. Math. 2007; link to arXiv though; see Conjecture A)

The maximum of $|\zeta(1/2 + it)|$ in the interval $[0,T]$ is $$\exp( (1+o(1)) \sqrt{\frac{1}{2} \log T \log \log T}) .$$

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There are reason to doubt that the size of $\zeta(s)$ alone could be responsible for the truth of the Riemann Hypothesis, however bounds for $\zeta(s)$ could be equivalent to a statement of the form "RH does not fail massively", as I will explain further.

Let me present one argument to convince you that the size of $\zeta(s)$ should be independent of the truth of RH: Suppose that the following (unlikely, but currently not ruled out) configurations of zeros occur in infinitely many intervals $[T; T + 1]$: we have roughly $\asymp \log T / \log\log T$ clusters of $\log\log T$ zeros, then in such interval $\zeta(s)$ should be of size $\exp(c \log T / \log\log T)$ (in particular contradicting the conjecture of Farmer, Gonek and Hughes). And then imagine that there are a few (say $4$) zeros of $\zeta(s)$ lying off the critical line. The two behaviors are envisage-able to occur simultaneously, unless of course we prove the falsehood/truth of each statement independently.

To wit: The size of $\zeta(s)$ is in a sense a local behavior, having $O(\log T)$ badly placed zeros in a $O(1)$ vicinity of a point $1/2 + it$ is enough to produce a very large (if not super large) value of $\zeta(s)$ at that point. Therefore the size of $\zeta(s)$ will not be affected by the truth or a small failure of the Riemann Hypothesis. However good bounds for $\zeta(s)$ can prevent the Riemann Hypothesis from failing badly. For example a result of Turan and Halasz asserts that if the Lindelof Hypothesis is true then there are at most $O(T^{\varepsilon})$ zeros in the half-plane $\sigma > \tfrac 34$.

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