Ramanujan's function \(k(\tau)=r(\tau)r^2(2\tau)\) and its modularity
The content of this paper is related to the following topics: Ramanujan's function $k(\tau)$, modular function, class field theory, congruence subgroup, Rogers-Ramanujan continued fraction, Kronecker's congruence. These topics have been used in analytic number theory. The authors provide the modularity of Ramanujan's function $k(\tau)$ associated with the Rogers-Ramanujan continued fraction. They present not only the modular equation of the function $k(\tau)$ with some symmetry relations and some congruence relations which are satisfied by the modular equations, but also for some $\tau$ in an imaginary quadratic field, the value $k(\tau)$ generates the ray class field over an imaginary quadratic field modulo 10 since the function $k$ is a generator of the field of the modular function on $\Gamma_1(10)$. The authors give an optimal way of evaluating the singular values of the function $k(\tau)$ with the aid of the modular equations associated with the $j(\tau)$ which denotes the elliptic modular function. For the details, fallow the content of the paper. Moreover, the authors provide some calculations with the aid of the MAPLE program.
scientific article; zbMATH DE number 7435586
scientific article