Three Ramanujan continued fractions with modularity

Yoonjin Lee, Yoon Kyung Park

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)299-323
Number of pages25
JournalJournal of Number Theory
Volume188
DOIs
StatePublished - Jul 2018

Keywords

  • Class field theory
  • Modular function
  • Ramanujan continued fraction

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