Euler's notebooks: The notes concerning analytical number theory, series and continued fractions (Q2713215)

Euler's heritage is so vast in significance and output, that it was hard to make even a bibliography of his works [\textit{G. Eneström}, Verzeichnis der Schriften Leonhard Eulers, Jahresber. Deutsch. Math.-Ver. Ergänzungsbände. Leipzig, IV Bd. L.1-2 (1910; JFM 41.0010.05 and JFM 44.0010.01)]. An initiative to publish \textit{Leonhardi Euleri opera omnia} came about only at the beginning of the twentieth century, to be a collaboration of the three countries closely related to Euler: Switzerland, Russia, and Germany. \textit{Opera omnia} included only published manuscripts, whereas everything else, like Euler's notebooks were slated for publication on a different occassion. Euler's notebooks were returned to the archives of the St. Petersburg's branch of the Russian Academy of Sciences, after World War II. The authors lead us through a part of these notebooks that reveal plenty about Euler's working and investigative methods that included, purely computational (experimental) aspects. The notebooks contain also Euler's catalogue (that he made) of his 539 books in Latin, German, French, English, Russian and Greek, in all areas of science and the humanities. NEWLINENEWLINENEWLINEIt was V. I. Smirnov who initiated the study of the notebooks in 1954. A work of a number of researchers culminated in 1997 with the publication of \textit{Unpublished materials of L. Euler}, edited by \textit{G. P. Matvievskaya} and \textit{E. P. Ozhigova} et al, see Zbl 0908.01040. The work in this area is not completed and the present paper lists some of the remaining unpublished results from Euler's notebooks. These problems are in NEWLINENEWLINENEWLINE1. Analytic number theory (Goldbach conjecture, distribution of primes, functional equations equivalent to the functional equation for the zeta function), NEWLINENEWLINENEWLINE2. Series (The Euler-Maclaurin summation formula and its applications, methods of summation of the series \(\sum_{i=1}^\infty i^{-2n}=\delta\pi^{2n}\) and their applications -- for instance, for \(n=4\), \(\delta=1/9450\), summation of series in correspondence with Goldbach, calculations of some transcendental quantities using series -- for instance he computes \(\pi\) to 128 places and its logarithms to 35 decimal places, divergent series, recurrent series, binomials, functional series, various other series), NEWLINENEWLINENEWLINE3. Continued fractions (relations with infinite series and products, relations with some transcendental quantities, applications of continued fractions to solutions of differential equations and integration of algebraic expressions, relations with quadratic equations and quadratic irrationalities, applications of continued fractions in solving number theoretic equations). NEWLINENEWLINENEWLINEThere are forty references and a number of misprints in this paper.