Sharpening of Lindelöf hypothesis - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:19:12Z http://mathoverflow.net/feeds/question/106503 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106503/sharpening-of-lindelof-hypothesis Sharpening of Lindelöf hypothesis Bazin 2012-09-06T12:06:04Z 2012-09-08T03:37:44Z <p>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. </p> <p>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 $(RH)$?</p> http://mathoverflow.net/questions/106503/sharpening-of-lindelof-hypothesis/106514#106514 Answer by Micah Milinovich for Sharpening of Lindelöf hypothesis Micah Milinovich 2012-09-06T14:24:29Z 2012-09-06T14:24:29Z <p>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. </p> <p>Ramachandra and Balasubramanian had previous proved an inequality of a similar form with the factor of $1-\varepsilon$ replaced by a smaller constant.</p> http://mathoverflow.net/questions/106503/sharpening-of-lindelof-hypothesis/106516#106516 Answer by quid for Sharpening of Lindelöf hypothesis quid 2012-09-06T14:42:36Z 2012-09-06T14:42:36Z <p>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, <a href="http://arxiv.org/abs/math/0506218" rel="nofollow">The maximum size of L-functions</a>, J reine angew. Math. 2007; link to arXiv though; see Conjecture A)</p> <p>The maximum of $|\zeta(1/2 + it)|$ in the interval $[0,T]$ is $$\exp( (1+o(1)) \sqrt{\frac{1}{2} \log T \log \log T}) .$$</p> http://mathoverflow.net/questions/106503/sharpening-of-lindelof-hypothesis/106643#106643 Answer by anonnn for Sharpening of Lindelöf hypothesis anonnn 2012-09-08T03:27:25Z 2012-09-08T03:37:44Z <p>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.</p> <p>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. </p> <p><strong>To wit</strong>: 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 $\sigma > \tfrac 34$. </p>