TITLE: Recent progress in the irrationality measure of $\pi$
QUESTION [12 upvotes]: The context.
For any real number $x$, let's define the quantity
$$\mu(x):=\sup\left\{\mu\in\mathbb R_+\, \text{there is an infinity of $(p,q)\in\mathbb Z\times\mathbb N$ such that}\ 0<\left\vert x-\frac pq\right\vert<\frac 1{q^{\mu}}\right\},$$
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.


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

Does this bound have been improved?

REPLY [8 votes]: 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.