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In the proof of Prop. 19 in these notes, Tao uses that for $|t|\leq 100$ and $\delta \ll |t|$ one has $$|\zeta(1+\delta + it)| \asymp \frac{1}{|t|}.$$ (i.e. in the region $\mathscr R:= \{s=\sigma+it : |t|\leq 100, 1 <\sigma -1 \leq m|t|\}$ for some "slope" $m$ visualized here on Desmos, there are "implicit constants" $c,C>0$ s.t. $$c |t|^{-1} \leq |\zeta(s)| \leq C |t|^{-1}.$$

I was able to prove this in a very "stupid" way. For $s\in B(1, \epsilon)$, the Laurent series expansion gives $\zeta(s) = \frac{1}{s-1} + O(1)$, so in $\mathscr R \cap B(1, \epsilon)$ we get the desired bounds. Then on the compact set $\overline{\mathscr R}\setminus B(1,\epsilon)$, we know that $\zeta$ is takes finite values (no poles), and is non-zero (in particular using the sledgehammer of nonvanishing of $\zeta$ on the line $\Re(s)=1$...), we get $|\zeta(s)| \asymp 1 \asymp |t|^{-1}$.

I am very displeased with this, because this proof really didn't say anything about the decay of the zeta function on vertical lines, which I think was the point of the asymptotic.

Here's my question:

Question: is it true that on vertical lines right of the critical strip (i.e. $\Re(s)=\sigma >1$), $$|\zeta(s)| \asymp_{\sigma} \langle t\rangle ^{-1}?$$ Notation: $\langle t\rangle:= (1+|t|^2)^{1/2}$ is the so-called Japanese bracket (which in particular is sim to $\asymp \min\{|t|^{-1}, 1\}$ if that is more intuitive notation), and the subscript means the implicit constant depends on $\sigma$.

If not, what about on truncated vertical lines $\{\sigma + it: |t|\leq T\}$; and then how does the implicit constant depend on $T$? How do the implicit constants behave depending on $\sigma$? For example, restricting to a compact/bounded range of $\sigma$ and/or $T$, do we get uniform implicit constants?

Remarks: there are a number a questions on the asymptotic behavior of zeta on vertical lines in the critical strip, like https://mathoverflow.net/questions/162625/leading-order-behaviour-of-riemann-zeta-function; the answers therein tell us that the universality of the zeta function means that there can be no such asymptotics in the critical strip (note that the universality of the zeta function is limited to inside the critical strip). There are of course some upper bounds Zeta function's asymptotic $\zeta(\sigma+it) = \mathcal{O}(|t|^{1-\sigma+\epsilon})$.

D.R.
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  • Look up the universality of the Riemann Zeta function. I think this'll dispute your first claim on the Japanese Bracket asymptotic behaviour. – Sean Nemetz Jun 06 '23 at 04:00
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    @SeanNemetz zeta function universality seems to only be relevant when talking about values inside the critical strip, at least from what I see on the Wikipedia page. – D.R. Jun 06 '23 at 04:05
  • oh my apologies, I misread the $1+\delta+it$ for $\frac{1}{2}+\delta+it$. – Sean Nemetz Jun 07 '23 at 01:03
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    See chapter $11$ of Titchmarsh RZ book for $\zeta$ on vertical lines; general Dirichlet series stuff implies that if $\sigma >1, \zeta(\sigma+it)$ takes or has as limits all allowable values $t \to \infty$ so, for example, there is $t_n \to \infty, \zeta(\sigma+it_n) \to \zeta(\sigma)$ which is, of course, the highest value possible in absolute value; similarly, we either can have $\zeta(\sigma+it_n) \to 0$ if $\sigma$ close enough to $1$ to allow it or $\zeta(\sigma+it_n) \to 2-\zeta(\sigma)$ if that is positive and then that is the lowest value allowable – Conrad Jun 07 '23 at 15:21

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On a fixed vertical line $\Re s=\sigma$ with $\sigma>1$, the average value of $\zeta(\sigma+it)$ is asymptotically $1$ (by absolute convergence one can average each term of the Dirichlet series separately). So there's definitely no way $\zeta(\sigma+it)$ can decay as $|t|\to\infty$.

Indeed $\zeta(\sigma+it)$ will have a limiting distribution on such a line, which is the same as the limiting distribution of the random variable $$ \prod_p (1-Z_p p^{-\sigma})^{-1}, $$ where the $Z_p$ are independent random variables (indexed by the primes) that are uniformly distributed on the unit circle in $\Bbb C$.

Greg Martin
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  • Thanks! Can you also provide a reference? And by "limiting distribution on such a line", do you mean the limit of the measures: cut up $\mathbb C$ into tiny boxes, and find the normalized measure of points on the vertical line segment $\sigma + i[-T,T]$ that land in a given box? And finally is it possible to give some sort of description of what the random variable distribution will look like (like a picture, some sort of "heat map" $\subseteq \mathbb C$ for where the probability distribution is most concentrated)? – D.R. Jun 06 '23 at 04:40