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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.

1930 edition

In the 1986 edition we have the following improvement: 1986 edition

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.

Shree
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  • not sure what you are talking about as Chapter 8 clearly shows that both $\zeta(1+it)$ and $1/\zeta(1+it)$ are $\Omega (\log \log t)$ for $t \to \infty$ so in particular they are unbounded and inferiorly can get as close to zero as wanted in absolute value – Conrad Oct 15 '20 at 14:18

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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.

reuns
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  • I see. Taking reciprocal of 14.9.1 we can get lower bound. The exact value doesn’t matter. – Shree Feb 19 '20 at 13:15
  • But 1939 edition is saying (essentially) that there exist infinity many large values of $t$ where $a \log \log t \le 1/|\zeta(1+it)| \le b \log \log t$ and 1986 edition is saying (at least on the attached page) $ \frac{1}{ a \log \log t} \le 1/ |\zeta(1+it)| \le b \log \log t$ 1986 edition does not rule out existence of fixed $M_1 $ and $M_2$ such that $ \frac{1}{ a \log \log t} \le M_1 \le 1/|\zeta(1+it) |\le M_2 \le b \log \log t $. This is impossible with 1939 version. – Shree Feb 19 '20 at 13:53