E352 -- Remarques sur un beau rapport entre les series des puissances tant directes que reciproques

(Remarks on a beautiful relation between direct as well as reciprocal power series)

Summary:

Euler evaluates the Riemann zeta function for some values and finds some functional relations for it.

Publication:

Originally published in Memoires de l'academie des sciences de Berlin 17, 1768, pp. 83-106
Opera Omnia: Series 1, Volume 15, pp. 70 - 90

Documents Available:

Original publication: E352
E352 can be viewed or downloaded from Digitalisierte Akademieschriften und Schriften zur Geschichte der Königlich Preußischen Akademie der Wissenschaften, which includes serial publications of the Prussian Academy of Science in the 18th and 19th Centuries.
A translation of E352 has been prepared by Thomas Osler and Lucas Willis. The translation is both faithful and highly readable, and is recommended.
Osler and Willis have also prepared a section-by-section Synopsis of E352. This is a good document to read for those looking for just the main ideas this work.
The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E352 include:
- Ricardo Pérez-Marco, "Eñe product in the transalgebraic class" (Dec 2019). Preprint available on Arxiv.org.
- Dutka J., "On the summation of some divergent series of Euler and the zeta functions." Archive for History of Exact Sciences, 50 (2), pp. 187-200 (1996).
- Ferraro G., "Differentials and differential coefficients in the Eulerian foundations of the calculus." Historia Mathematica, 31 (1), pp. 34-61 (Feb 2004).
- Ferraro G, Panza, M., "Developing into series and returning from series: A note on the foundations of eighteenth-century analysis." Historia Mathematica, 30 (1), pp. 17-46 (Feb 2003).
- Riemann used this article in 1859 when he wrote "Über die Anzahl der Primzahlen unter einer gegebenen Gröbe".

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