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 -