Eisenstein series and approximations to \(\pi\) (Q5954253)

The first purpose of this paper is to elucidate some marginal notes of Ramanujan found in his ``lost notebook''. It turns out that these notes give relations between certain values of \(Q^3\) and \(R^2\) where \(Q\) and \(R\) are Ramanujan's usual notations for the Eisenstein series of weight 4 and 6. The authors discover that the values are at \(q=-\exp (-\pi\sqrt n)\) with \(n=11\), 19, 27, 43, 67, 163 and 51. The authors first deduce Ramanujan's relations from known values of modular functions at singular moduli. Having done this they introduce Ramanujan's function \(P\), which is not quite a modular form of weight 2. They introduce two sequences \(a_n\) and \(b_n\) which depend on the values \(P,Q\) and \(R\) at the points given above. These lead, at least in the cases above, to relations of the formNEWLINE\[NEWLINEQ^{-1/2} \left(\sqrt {\text{n.P}}-{6\over\pi} \right)= \text{explicit algebraic number}NEWLINE\]NEWLINE where \(P\) and \(Q\) are evaluated at \(-\exp(-\pi \sqrt n)\). One can also regard these formulae, when combined with the series expansions of \(P\) and \(Q\), as giving approximations to \(\pi\). In the final two sections the authors use the differential equation satisfied by \(P\) and the modular equation for \(j\) to develop a self-contained proof of this result. Again they discuss several numerical examples. They also discuss the relationship of their results to those of the Borweins and the Chudnovskys.