A level 16 analogue of Ramanujan series for 1/π

Abstract

The modular function h(τ)=q∏n=1∞[Formula presented] is called a level 16 analogue of Ramanujan's series for 1/π. We prove that h(τ) generates the field of modular functions on Γ0(16) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K(h(τ)) modulo 4 over an imaginary quadratic field K for τ∈K∩H such that Z[4τ] is the integral closure of Z in K, where H is the complex upper half plane. For any τ∈K∩H, it turns out that the value 1/h(τ) is integral, and we can also explicitly evaluate the values of h(τ) if the discriminant of K is divisible by 4. © 2017 Elsevier Inc.