Three Ramanujan continued fractions with modularity
- 주제(키워드) Class field theory , Modular function , Ramanujan continued fraction
- 관리정보기술 faculty
- 등재 SCIE, SCOPUS
- 발행기관 Academic Press Inc.
- 발행년도 2018
- URI http://www.dcollection.net/handler/ewha/000000151631
- 본문언어 영어
- Published As http://dx.doi.org/10.1016/j.jnt.2018.01.012
초록/요약
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. © 2018 Elsevier Inc.
more