Continued Fractions and Irrationality Measures for Chowla-Selberg Gamma Quotients

Abstract: We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(\Gamma(1/3)/\Gamma(2/3))^3$, for $a^{1/4}\operatorname{CS}(-4)=a^{1/4}(\Gamma(1/4)/\Gamma(3/4))^2$ with $a=12$ and $a=1/5$, and for $\operatorname{CS}(-7)=\Gamma(1/7)\Gamma(2/7)\Gamma(4/7)/(\Gamma(3/7)\Gamma(5/7)\Gamma(6/7))$. These appear to be the first proved and reasonable irrationality measures for gamma quotients.

dynamical systemsnumber theory

Audience: researchers in the topic

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