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**Ramanujan: a tale of two evaluations.**
*(English)*
Zbl 1352.11088

Summary: 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.

B. C. Berndt has done a remarkable job of editing the notebooks Ramanujan left behind. In particular, Berndt’s Chapter 9 of [Ramanujan’s notebooks. Part I. New York, etc.: Springer-Verlag (1985; Zbl 0555.10001)] 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” [5, page 233] 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.

B. C. Berndt has done a remarkable job of editing the notebooks Ramanujan left behind. In particular, Berndt’s Chapter 9 of [Ramanujan’s notebooks. Part I. New York, etc.: Springer-Verlag (1985; Zbl 0555.10001)] 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” [5, page 233] 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.

### MSC:

11M99 | Zeta and \(L\)-functions: analytic theory |

11G55 | Polylogarithms and relations with \(K\)-theory |

40G99 | Special methods of summability |

### Citations:

Zbl 0555.10001
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\textit{D. J. Manzoli}, Rocky Mt. J. Math. 46, No. 3, 925--938 (2016; Zbl 1352.11088)

### References:

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[5] | B.C. Berndt, Ramanujan’s notebooks Part I, Springer-Verlag, New York, 1985. · Zbl 0555.10001 |

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[9] | L. Lewin, Polylogarithms and associated functions , North Holland, New York, 1981. · Zbl 0465.33001 |

[10] | —-, ed., Structural properties of polylogarithms , American Mathematical Society, Providence, RI, 1991. · Zbl 0745.33009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.