## Abstract

We study the modularity of Ramanujan's function k (τ) = r (τ) r2 (2 τ), where r (τ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k (τ) of "an"level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ in an imaginary quadratic field, the value k (τ) k generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 (10). Furthermore, we suggest a rather optimal way of evaluating the singular values of k (τ) using the modular equations in the following two ways: one is that if j (τ) is the elliptic modular function, then one can explicitly evaluate the value k (τ) and the other is that once the value k (τ) is given, we can obtain the value k (r τ) for any positive rational number r immediately.

Original language | English |
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Pages (from-to) | 1727-1741 |

Number of pages | 15 |

Journal | Open Mathematics |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

- Ramanujan's function k
- class field theory
- modular function

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