# On the Absolute Value of the Riemann Zeta Function on the Critical Line [closed]

Is there any sort of (closed form preferably, though if not, it's fine) function for $$|\zeta(\frac12+it)|$$ where $$\zeta$$ is the Riemann zeta function? Anything is welcome, so I can take it from there. Upper bounds, lower bounds and asymptotics are also accepted.

• How about the Riemann Siegel formula?
– user156885
May 5, 2020 at 6:50
• @zz7948 I'm not sure how that helps. May 5, 2020 at 16:19
• @Dave related to approximating zeta. Good for numerical approximations and maybe the best we have to a ’closed form’ in some situations. Taking it with the ’anything is welcome’ attitude
– user156885
May 5, 2020 at 16:54
• @zz7948 Yeah, but that was for zeta, not for the absolute value of it. May 5, 2020 at 17:23

It is not known whether, for $$t>0$$ large, $$|\zeta(1/2+it)|$$ is smaller than $$t^{1/7}$$, while it is conjectured that $$1/7$$ can be replaced by any positive exponent. This should answer your question. For more information, consult Titchmarsh's book on the Riemann zeta function, there is a whole chapter on this topic. Another good starting point is the relevant Wikipedia entry.
• Though when they say "$c_{k,j}$ as some constant", what kind of constants are they talking about? May 5, 2020 at 3:41
• @QuoteDave: That asymptotic formula is only conjectured. As the Wikipedia page says, a conjectured formula for $c_{k,j}$ was given by Keating & Snaith (2000). The best unconditional upper bounds appear in the table in the Wikipedia page. Please study the literature. It is vast and easy to find. May 5, 2020 at 3:46
• No, I am really only interested in $k=1$, which has been proven, and the Wikipedia page never gave any values of $c$. May 5, 2020 at 3:49