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The context.

For any real number $x$, let's define the quantity

$$\mu(x):=\inf\left\{\mu\in\mathbb R_+\, \text{there is an infinity of rationals $p/q$ such that}\ \left\vert x-\frac pq\right\vert<\frac 1{q^{\mu}}\right\},$$

and let's call it the irrationality measure of $x$.

We know that

$$\mu(\pi)\leqslant 8.016$$

thanks to M. Hata (1992).

The question.

We can read on this Wikipedia page that

\begin{equation} \mu(\pi)\leqslant 7.6063, \end{equation}

but this is provided without any reference.

  1. Do you know any article where I could find a mention of this result?

  2. Does this bound have been improved?

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    $\begingroup$ Mathworld has a reference for the smaller bound: "Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008." I don't know if there's been any progress since 2008. $\endgroup$ – Micah May 12 '18 at 13:23
  • $\begingroup$ @Micah That looks like a potential answer to the question to me. $\endgroup$ – Mark Bennet May 12 '18 at 19:35
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    $\begingroup$ @Micah: I don't believe a better value is currently known, because if there were a better value, then I'm sure it would turn up in the google search Salikhov + "on the irrationality measure of" (corresponding google scholar search). Also, this is the best value mentioned in Marko Leinonen's 2017 Ph.D. dissertation On Various Irrationality Measures. $\endgroup$ – Dave L. Renfro May 13 '18 at 17:55
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    $\begingroup$ @DaveL.Renfro: Thanks! That sounds good enough for an answer to me. $\endgroup$ – Micah May 13 '18 at 18:04
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Salikhov proved the smaller bound in: "Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008." as referenced, e.g., on Mathworld. This appears to be the best bound in the peer-reviewed literature at the moment (as of February 2020).

Zeilberger and Zudlin have a (not-yet-peer-reviewed) arXiv preprint in which they improve Salikhov's bound to $7.1032\dots$, here.

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  • $\begingroup$ It is now superseded by 7.103205334137..., according to mathworld and this paper: arxiv.org/abs/1912.06345 $\endgroup$ – George Lowther Feb 10 '20 at 18:15
  • $\begingroup$ @GeorgeLowther: Fantastic, thanks! $\endgroup$ – Micah Feb 10 '20 at 20:04

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