Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896931 | Journal of Number Theory | 2018 | 22 Pages |
Abstract
We study three Ramanujan continued fractions c(Ï),W(Ï) and T(Ï). In fact, c(Ï) and W(Ï) are modular functions of level 16, and T(Ï) is a modular function of level 32. We first prove that the values of c(Ï) and W(Ï) can generate the ray class field modulo 4 over an imaginary quadratic field K. We also prove that 2/(1âc(Ï)),1/W(Ï),T(Ï)+1/T(Ï) are algebraic integers for any imaginary quadratic quantity Ï. Furthermore, we find the modular equations of c(Ï),T(Ï) and W(Ï) for any level, and we show that c(Ï) and W(Ï) satisfy the Kronecker's congruence. We can express the value c(rÏ) (respectively, T(rÏ),W(rÏ)) in terms of radicals for any positive rational number r when the value c(Ï) (respectively, T(Ï),W(Ï)) can be written as radicals.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Yoonjin Lee, Yoon Kyung Park,