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Let $\zeta$ be the Riemann zeta function.

My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?

My searches in relevant texts like Titschmarsh reveal that people seem to be focused on the case $\sigma= 1/2$, for which Lindelof conjectured that $|\zeta(1/2+it)|\ll t^{\epsilon}$ for any $\epsilon>0$.

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I answer here for what is known for general $\sigma$, without assuming anything (in particular not LH). Let $s = \sigma + it$ where $s$ and $t$ are real numbers.

If $\mathbf{\sigma > 1}$, then the absolute convergence of the Dirichlet series yields $$\zeta(s) \ll 1 \qquad(\sigma>1)$$ (here and below, the $\ll$-constant may depend on $\sigma$.

If $\mathbf{\sigma < 0}$, then we can use the functional equation in order to come back to the known region $\sigma > 1$. Indeed, recall that $\zeta(s) = \gamma(s) \zeta(1-s)$ for a certain explicit completing factor $\gamma(s)$. It can be written in terms of $\Gamma$ functions and the Stirling formula easily leads to $$\gamma(s) \ll_\varepsilon |t|^{\frac{1}{2} - \sigma + \varepsilon}$$ Since $\zeta(1-s)$ is bounded for $\sigma < 0$ by the previous case, we deduce $$\zeta(s) \ll |t|^{\frac{1}{2} - \sigma} \qquad(\sigma<0)$$

If $\mathbf{0 \leqslant \sigma \leqslant 1}$, the Phragmén-Lindelöf principle allows to interpolate the bounds above in the critical strip, so that $$\zeta(s) \ll |t|^{\frac{1}{2}(1 - \sigma)}\qquad(0<\sigma<1)$$

Good references for such matters are for instance Montgomery-Vaughan, Multiplicative Number Theory, 1. Classical Theory, or Tenenbaum, Introduction to Analytic and Probabilistic Number Theory.

As for the Lindelöf hypothesis (implied by RH, so this bound holds under RH as well.), it states as you said that for $\sigma = 1/2$ a certain bound is satisfied, namely $$\zeta(s) \ll_\varepsilon |t|^\varepsilon.$$

Is you are interested in what happens in the whole critical strip, you can apply the Phragmén-Lindelöf principle as well to sharpen the bound. You get assuming LH (or RH), \begin{align*} \zeta(s) &\ll_\varepsilon |t|^{\varepsilon}\qquad\left(\frac12\le\sigma<1\right) \\ \zeta(s) &\ll_\varepsilon |t|^{\frac12-\sigma+\varepsilon}\qquad\left(0<\sigma<\frac12\right) \end{align*}

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  • $\begingroup$ Thanks, but i cannot find any of those texts on the web. In particular, for some $\sigma<1/2$, can $|\zeta(\sigma+it)|$ be as large as $t^{\theta}$ for infinitely many $t$, where $\theta$ is some positive constant ? $\endgroup$ – etihad10 Dec 27 '18 at 0:48
  • $\begingroup$ @etihad10 I just edited my answer to add what can be said under LH for bounds on vertical lines other than the critical line, as an application of the Phragmén-Lindelöf principle. $\endgroup$ – Desiderius Severus Dec 27 '18 at 0:52
  • $\begingroup$ Sorry if i'm missing something, but you stated that $\zeta(s)\ll _{\epsilon}|t|^{2(1-\sigma)\epsilon}$ presumably for every $\epsilon>0$, doesn't this entail the LH ? $\endgroup$ – etihad10 Dec 27 '18 at 1:03
  • $\begingroup$ @etihad10 This bound is indeed deduced from Lindelöf Hypothesis, that was to make a clear answer to your question: no matter what bounds you use on some vertical lines (either known results as in the first part of my answer, or LH as in the last two displayed formulas), you can deduce bounds for the vertical line $\sigma < 1/2$. $\endgroup$ – Desiderius Severus Dec 27 '18 at 1:23
  • $\begingroup$ Thanks, so in summary you are saying for any $\sigma \in (0,1)$, the LH entails $|\zeta(\sigma+it)|\ll _{\epsilon}|t|^{\epsilon}$ for every $\epsilon>0$, right ? $\endgroup$ – etihad10 Dec 27 '18 at 1:40

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