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Going through what it says on page 95 of Titchmarsh's book on the Zeta function, and using his remark about $\mu$ at the bottom of that page, I conclude that there is some constant $C>0$ so that: $$\left|\zeta\left(it\right)\right|\leq C\sqrt{\left|t\right|},\textrm{ }\forall t\in\mathbb{R}$$

In my current work, I need to know an explicit value for this $C$. Graphing the quotient: $$\frac{\left|\zeta\left(2^{t}i\right)\right|}{\sqrt{2^{t}}}$$ gives an apparent global maximum of slightly less than 1.37, occurring at $t≈8.9082$

Is this rigorously justifiable? I.e., is:

$$\left|\zeta\left(it\right)\right|\leq 1.37\sqrt{\left|t\right|},\textrm{ }\forall t\in\mathbb{R}$$

true? More generally, what's the most accurate estimate of this type which is currently known (with explicit constants)?

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  • $\begingroup$ You need to read the definition of $\mu(0)$ more closely. It is the (greatest) lower bound of the numbers $\xi$ such that $|\zeta(it)|$ is $O(t^\xi)$. Presumably the implicit constant in each Big Oh can depend on $\xi$, and even grow as $\xi$ decreases. Even though $\mu(0)=1/2$, this does not prove $\zeta(it)=O(t^{1/2})$. (The Big Oh may be true, but is does not follow from what's on p.95.) $\endgroup$
    – Stopple
    Commented Feb 14, 2020 at 16:23
  • $\begingroup$ I was worried about that. Darn. $\endgroup$
    – MCS
    Commented Feb 14, 2020 at 20:29

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