## Abstract

We study three Ramanujan continued fractions c(τ),W(τ) and T(τ). In fact, c(τ) and W(τ) are modular functions of level 16, and T(τ) is a modular function of level 32. We first prove that the values of c(τ) and W(τ) can generate the ray class field modulo 4 over an imaginary quadratic field K. We also prove that 2/(1−c(τ)),1/W(τ),T(τ)+1/T(τ) are algebraic integers for any imaginary quadratic quantity τ. Furthermore, we find the modular equations of c(τ),T(τ) and W(τ) for any level, and we show that c(τ) and W(τ) satisfy the Kronecker's congruence. We can express the value c(rτ) (respectively, T(rτ),W(rτ)) in terms of radicals for any positive rational number r when the value c(τ) (respectively, T(τ),W(τ)) can be written as radicals.

Original language | English |
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Pages (from-to) | 299-323 |

Number of pages | 25 |

Journal | Journal of Number Theory |

Volume | 188 |

DOIs | |

State | Published - Jul 2018 |

### Bibliographical note

Funding Information:The authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ), the first author was also supported by the National Research Foundation of Korea ( NRF-2017R1A2B2004574 ) and the second-named author was also supported by the National Research Foundation of Korea ( NRF-2017R1D1A1B03029519 ).

Funding Information:

The authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827), the first author was also supported by the National Research Foundation of Korea (NRF-2017R1A2B2004574) and the second-named author was also supported by the National Research Foundation of Korea (NRF-2017R1D1A1B03029519).

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Class field theory
- Modular function
- Ramanujan continued fraction