Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients
Henri Cohen, Wadim Zudilin
TL;DR
The paper develops a framework to produce explicit irrationality measures for Chowla--Selberg gamma quotients $ ext{CS}(D)$ by constructing rapidly convergent continued fractions tied to CM-values of modular functions. It provides a central CF family, three complementary ways to compute the CF limits (Laguerre-based, hypergeometric, and elliptic-integral expressions), and a modular-hypergeometric machinery that connects CM points to gamma quotients via Hauptmoduln. By carefully bounding denominators through $d_D(n)$ and $d_D^*(n)$ and applying a standard irrationality-criterion, the authors obtain 20 proved irrationality measures, including for $ ext{CS}(-3)$ and several other negative discriminants; these are among the first rigorous irrationality measures for gamma quotients. The work also maps a path to generalizations involving cocompact triangle groups and Shimura curves, with potential links to other hypergeometric and Ramanujan-type evaluations, thereby blending modular forms, CM theory, and Diophantine approximation in a novel way.
Abstract
We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(Γ(1/3)/Γ(2/3))^3$, for $a^{1/4}\operatorname{CS}(-4)=a^{1/4}(Γ(1/4)/Γ(3/4))^2$ with $a=12$ and $a=1/5$, and for $\operatorname{CS}(-7)=Γ(1/7)Γ(2/7)Γ(4/7)/(Γ(3/7)Γ(5/7)Γ(6/7))$. These appear to be the first proved and reasonable irrationality measures for gamma quotients.