2024/02/13 by John M. Campbell, Campbell, John M.
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2402.08485
Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number h(-d) = 1 for a quadratic form with discriminant d = 163. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky-Chudnovsky formula and the Heegner number d = 163 such that ℚ( √(-d) ) has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for λ∗(163) and α(163), which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for \frac1π generalizing the Chudnovsky algorithm.