Sharpening of Lindelöf hypothesis - MathOverflow

Sharpening of Lindelöf hypothesis

2012-09-06T12:06:04Z

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 $(RH)$?

Answer by Micah Milinovich for Sharpening of Lindelöf hypothesis

2012-09-06T14:24:29Z

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.

Answer by quid for Sharpening of Lindelöf hypothesis

2012-09-06T14:42:36Z

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

Answer by anonnn for Sharpening of Lindelöf hypothesis

2012-09-08T03:27:25Z

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 $\sigma > \tfrac 34$.