The irrationality measure of $\pi$

Description of constant

We define $C_{7a}$ to be the irrationality measure of $\pi$:

$C_{7a} := \sup_{\mu\in\mathbb{R}} \mu$ such that $\lvert \pi - p/q \rvert < q^{-\mu}$ for infinitely many rationals $p/q$.

Equivalently, $C_{7a}$ is the infimum of all $\nu$ such that for every $\varepsilon>0$ there exists $q_{0}(\varepsilon)$ with

\[\left|\pi-\frac{p}{q}\right| > \frac{1}{q^{\nu+\varepsilon}}\]

for all integers $p$ and all integers $q \ge q_{0}(\varepsilon)$.

Known upper bounds

Bound Reference Comments
$42$ [M1953] First proof that $C_{7a}$ is finite (i.e. $\pi$ is not a Liouville number).
$20.6$ [Mi1974] Improves Mahler’s exponent.
$19.8899944$ [C1982] Uses Hermite–Padé approximation methods.
$14.797074$ [RV1993] Follows from an effective irrationality measure for $\zeta(2)=\pi^2/6$.
$8.016045$ [H1993] Record for many years (Hata gave a series of improvements).
$7.606308$ [S2008] Salikhov’s bound.
$7.103205334137$ [ZZ2020] Current record bound.

Known lower bounds

Bound Reference Comments
$2$ [D1842] Dirichlet’s theorem; holds for every irrational number (in particular for $\pi$).

References