Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $(RH)$?

share|improve this question
    
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. –  Bazin Sep 6 '12 at 17:21
    
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. –  Bazin Sep 6 '12 at 17:23
1  
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. –  quid Sep 6 '12 at 18:17
    
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). –  GH from MO Sep 7 '12 at 2:17
add comment

3 Answers

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.

share|improve this answer
add comment

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 $$\exp( (1+o(1)) \sqrt{\frac{1}{2} \log T \log \log T}) .$$

share|improve this answer
add comment

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 $\sigma > \tfrac 34$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.