Rocky Mountain Journal of Mathematics

Ramanujan: A tale of two evaluations

Donald J. Manzoli

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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.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 3 (2016), 925-938.

Dates
First available in Project Euclid: 7 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1473275767

Digital Object Identifier
doi:10.1216/RMJ-2016-46-3-925

Mathematical Reviews number (MathSciNet)
MR3544840

Zentralblatt MATH identifier
1352.11088

Subjects
Primary: 40G99: None of the above, but in this section

Keywords
Ramanujan polylogarithms

Citation

Manzoli, Donald J. Ramanujan: A tale of two evaluations. Rocky Mountain J. Math. 46 (2016), no. 3, 925--938. doi:10.1216/RMJ-2016-46-3-925. https://projecteuclid.org/euclid.rmjm/1473275767


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