TY - JOUR
T1 - Three Ramanujan continued fractions with modularity
AU - Lee, Yoonjin
AU - Park, Yoon Kyung
N1 - 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.
PY - 2018/7
Y1 - 2018/7
N2 - 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.
AB - 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.
KW - Class field theory
KW - Modular function
KW - Ramanujan continued fraction
UR - http://www.scopus.com/inward/record.url?scp=85042850623&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2018.01.012
DO - 10.1016/j.jnt.2018.01.012
M3 - Article
AN - SCOPUS:85042850623
VL - 188
SP - 299
EP - 323
JO - Journal of Number Theory
JF - Journal of Number Theory
SN - 0022-314X
ER -