On some new explicit evaluations of class invariants
The Ramanujan-Weber class invariants \(G_n\) and \(g_n\) are \[ G_n := 2^{-1/4}q^{-1/24}\chi(q) \qquad\text{ and }\qquad g_n := 2^{-1/4}q^{-1/24}\chi(-q), \] where \(\chi(q) = (1+q)(1+q^3)(1+q^5)\cdots \) and \( q = e^{-\pi \sqrt{n}}. \) A number of values of \(G_n\) and \(g_n\) are given in this paper. Specifically, \(G_n\) is evaluated for \[ n = 30,10/3,42,14/3,78,26/3,102,34/3, \] while \(g_n\) is evaluated for \[ \begin{gathered} n = 15,5/3,21,7/3,33,11/3,39,13/3,57,19/3,93,31/3,177,59/3, \\ 60,20/3,84,28/3,120,40/3,132,44/3,156,52/3,168,56/3, \\ 228,76/3,312,104/3, 372,124/3,408,136/3. \end{gathered} \] The evaluations of \(G_n\), and \(g_n\) with \(n\) odd, are obtained using modular equations, other known class invariant evaluations, and relations involving \(G_n\), \(G_{9n}\), \(g_n\), \(g_{4n}\), \(g_{9n}\), and \(g_{36n}\). Details are given for \(G_{30}\), \(G_{10/3}\), \(g_{15}\) and \(g_{5/3}\), and the remaining cases are left to the reader. The evaluations of \(g_n\) with \(n\) even follow from the fact that \(g_{4n} = 2^{1/4}G_ng_n\). Values of the singular modulus \(\alpha_n\) (whose definition we do not give here, but which is equal to \((G_ng_{4n})^{-8}\)) are also recorded, for \[ \begin{gathered} n = 15,5/3,21,7/3,30,10/3,33,11/3,39,13/3,42,14/3, \\ 57,19/3,78,26/3,93,31/3,102,34/3.\end{gathered} \] The reader should be alerted that it will be necessary to consult Section 34 of [\textit{B. C. Berndt}, Ramanujan's notebooks. Part V. New York, NY: Springer (1998; Zbl 0886.11001)] for the known class invariant evaluations required for those proofs which are omitted. It might also be pointed out that in the authors' remark on p. \(123\), the identities numbered \((6.x)\) should be \((7.x)\).
scientific article; zbMATH DE number 5310220