Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909893824348160 |
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| author | Cohen, Henri Zudilin, Wadim |
| author_facet | Cohen, Henri Zudilin, Wadim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients Cohen, Henri Zudilin, Wadim Number Theory Classical Analysis and ODEs 11F11, 11F67, 11G15, 11J70, 11J82, 33C05, 33C45 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. |
| title | Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients |
| topic | Number Theory Classical Analysis and ODEs 11F11, 11F67, 11G15, 11J70, 11J82, 33C05, 33C45 |
| url | https://arxiv.org/abs/2510.00215 |