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2016 Ramanujan: A tale of two evaluations
Donald J. Manzoli
Rocky Mountain J. Math. 46(3): 925-938 (2016). DOI: 10.1216/RMJ-2016-46-3-925

Abstract

In 1887, beneath a canopy of stars, Srinivasa Ramanujan commenced his brief existence on this planet. In a universe at the mercy of its entropy, against all odds, a genius was born. His destiny was mathematics, a subject born thousands of years earlier. The power of this discipline is not to be denied. After all, with our minds in the stars, we have placed footprints on the moon. The conquest of the moon was a triumph of applied mathematics; however, it was the landscape of pure mathematics that awaited Ramanujan. In time, he would explore it with passion, leaving footprints lasting for eternity.

Professor Bruce C.~Berndt has done a remarkable job of editing the notebooks Ramanujan left behind. In particular, Berndt's Chapter 9 of \textit {Ramanujan's notebooks Part} I provides a magnificent in-depth look at raw mathematical talent in action. The primary purpose of this article is to present three Chapter~9 related results, a series evaluation and two new functional equations, that Ramanujan either missed or his work on them was lost. The secondary purpose is to present what Berndt calls a ``corrected version'' \cite [page 233]{Berndt} of an incorrect Chapter~9 formula of Ramanujan. This series evaluation represents one of the few serious mistakes to be found in Ramanujan's work.

Citation

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Donald J. Manzoli. "Ramanujan: A tale of two evaluations." Rocky Mountain J. Math. 46 (3) 925 - 938, 2016. https://doi.org/10.1216/RMJ-2016-46-3-925

Information

Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 1352.11088
MathSciNet: MR3544840
Digital Object Identifier: 10.1216/RMJ-2016-46-3-925

Subjects:
Primary: 40G99

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

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