Œuvres complètes. Collected papers. Ed. by Gerrit van Dijk. Volumes 1 and 2. (French, Dutch, English, German) Zbl 0779.01010

Stieltjes, Thomas Jan

Berlin: Springer-Verlag (ISBN 3-540-55560-9/hbk). ix, 566 p./v.1; vi, 750 p./v.2 (1993).

This is mainly a reissue of the Œuvres complètes of Stieltjes (1856–1894) as published by Noordhoff, 1914/18 (see JFM 45.0025.05 and JFM 46.0002.01), to which are added a short biography (van Dijk), four commentaries on significant parts of Stieltjes’s work, and an English translation of his most influential paper, “Recherches sur les fractions continues” (1894; JFM 25.0326.01). W. van Assche reports on continued fractions, orthogonal polynomials, moment problems, and Markov-Stieltjes inequalities, supplying a bibliography of 123 titles. F. Beukers describes how Stieltjes’s papers on number theory fit in with the historical development of that field, and are not confined to his attempts on the Riemann hypothesis. An article by W. A. J. Luxemburg on the Stieltjes integral shows how this concept, though already known to Cauchy, emerged from the work on continued fractions, and influenced the further development of integration and functional analysis. Stieltjes’s work on \(1/\zeta(s)\) which is connected to the famous conjecture of Mertens (1897) and later developments in analytic number theory are reported by H. J. J. te Riele.
In addition to the subjects of these commentaries some of Stieltjes’s earlier papers will be of interest to the historian. As an autodidact he appears to have studied writings of the early nineteenth century that were considered outdated by his contemporaries, and he drew much inspiration from those readings. He gave contributions to the foundations of real analysis, in particular differentiability and divergent series, and a lucid consideration of Fourier’s use of divergent series for the determination of Fourier coefficients. The only shortcoming of the present edition is that it fails to recognize these highly original aspects of the intellectual development of Stieltjes.

Reviewer: Detlef Laugwitz (Darmstadt)

MSC:

01A75	Collected or selected works; reprintings or translations of classics
01A55	History of mathematics in the 19th century
11-03	History of number theory
26-03	History of real functions
30-03	History of functions of a complex variable
40-03	History of sequences, series, summability

Keywords:

continued fractions; integration; analytic number theory; moment problem

Citations:

JFM 45.0025.05; JFM 46.0002.01; JFM 25.0326.01