Unboundedness of $\frac{1}{\zeta(1+it)}$ - difference between two versions of Titchmarsh 1930 and 1986

Question

In The Zeta Function if Riemann 1930 edition we see the following bounds (upper and lower) on $\limsup |\frac{1}{\zeta(1+it)}|$ (effectively)

*Note: I have accidentally highlighted the wrong equation below, the first equation is what was proven (in 1930) and second equation I highlighted is best case scenario and depends on a speculation.

In the 1986 edition we have the following improvement:

I’m wondering if anyone has insight into what happened to the lower bound in the latest edition. Was it (a) the state of knowledge in 1930 which got corrected in 1986 - which is likely case, (b) is there an omission in the latest edition, or (c) perhaps it is included somewhere in the 1986 edition and I haven’t found it.

If it is (a), then I am curious about who proved it wrong (eg was it a result of disproven conjecture)?

Thanks in advance.

Answer 1

Up to a constant the two are the same and nobody cares of the constant.

Under the RH $$|\zeta(1+it)| \in [a,b] \log \log t,\qquad t> 1$$

The simplest derivation doesn't give the optimal constant, and there are no much good reason to compute some constant explicitly, we are happy to just prove some exists. As usual with the RH, there is a theorem saying that whatever the constant is, if the zeros $\le 10^{18}$ are on the critical line then the bound is achieved for $t\le ...$ even if the RH is false.