The irrationality measure of Γ(1/4)

Description of constant

For a real number $\gamma$, its irrationality exponent $\mu (\gamma )$ is defined by $$\mu (\gamma ):=\inf \{c\in \mathbb{R}:|\gamma -\frac{a}{b}|\le |b{|}^{-c}\text{has only finitely many solutions }(a,b)\in {\mathbb{Z}}^2\}.$$

<a href="#Zud2004-def-mu">[Zud2004-def-mu]</a>

We define

$$C_{7b}:=\mu (\Gamma (1/4)).$$

For $p/q\in \mathbb{Q}$ in lowest terms with $q>0$, write

$$h(p/q):=\log \max (|p|,q).$$

This agrees with the absolute logarithmic height used by Bruiltet. <a href="#Suk2013-def-hpq">[Suk2013-def-hpq]</a> <a href="#Bru2002-def-h">[Bru2002-def-h]</a> <a href="#Bru2002-def-places">[Bru2002-def-places]</a>

Known upper bounds

Known lower bounds

Additional comments and links

• The large gap between the proven upper bound $10^{143}$ and the universal lower bound $2$ reflects the current weakness of methods for proving sharp irrationality measures for special constants such as $\Gamma (1/4)$.

• Wikipedia page on irrationality measure

References

• <a id="Bru2002"></a>[Bru2002] Bruiltet, Sylvain. D’une mesure d’approximation simultanée à une mesure d’irrationalité : le cas de $\Gamma (1/4)$ et $\Gamma (1/3)$. Acta Arithmetica 104 (2002), no. 3, 243–281. DOI: 10.4064/aa104-3-3. PDF. Google Scholar
• <a id="Bru2002-cor-gamma14"></a>[Bru2002-cor-gamma14] quote: “$h(p/q)\ge 10^{75}\Rightarrow |\Gamma (1/4)-p/q|>(1/(qe){)}^{10^{143}}$.”

• <a id="Bru2002-def-h"></a>[Bru2002-def-h] quote: “Pour $x=(x_1,\dots ,x_m)\in {\mathbb{Q}}^m$ on d´esigne par $\Vert x{\Vert }_v=\max (1,|x_1{|}_v,\dots ,|x_m{|}_v)$, puis $H(x)={\prod }_v\Vert x{\Vert }_v^{d_v}$ et $h(x)=\frac{\log H(x)}{d(x)}$ les hauteurs relative et absolue de $x$.”

• <a id="Bru2002-def-places"></a>[Bru2002-def-places] quote: “Si $K$ est un corps de nombres et $v$ une place de $K$, $|\cdot {|}_v$ d´esignera la valeur absolue normalis´ee associ´ee; on notera $d_v$ le degr´e local de $K$ en $v$.”

• <a id="Zud2004"></a>[Zud2004] Zudilin, Wadim. An essay on irrationality measures of $\pi$ and other logarithms. Preprint (2004). Google Scholar. arXiv PDF. arXiv abstract
• <a id="Zud2004-def-mu"></a>[Zud2004-def-mu] quote: “$$\begin{gathered} \mu (\gamma )=\inf \\ c\in \mathbb{R}:|\gamma -a/b|\le |b{|}^{-c}\text{has finitely many solutions} \end{gathered}$$.”

• <a id="CZ2025"></a>[CZ2025] Cohen, Henri; Zudilin, Wadim. Continued Fractions and Irrationality Measures for Chowla–Selberg Gamma Quotients. Preprint (2025). Google Scholar. arXiv PDF.

• <a id="Suk2013"></a>[Suk2013] Sukiennik, Justin. Bounds on Height Functions. Conference handout, 2013 Maine–Quebec Number Theory Conference (October 6, 2013). PDF
• <a id="Suk2013-def-hpq"></a>[Suk2013-def-hpq] quote: “When $K=\mathbb{Q}$, the height for $x/y\in \mathbb{Q}$ (in lowest terms) is described by $$\begin{gathered} h(x/y)=\log \max \\ |x|,|y| \end{gathered}$$.”

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.