Historical face of number theory(ists) at the turn of the 19th century. (English) Zbl 1370.01014
Steuding, Jörn (ed.), Diophantine analysis. Course notes from a summer school, Würzburg, Germany, July 21–24, 2014. Basel: Birkhäuser/Springer (ISBN 978-3-319-48816-5/hbk; 978-3-319-48817-2/ebook). Trends in Mathematics, 175-229 (2016).
The text consists of four parts:
Part 1 (“Meeting the Hurwitz brothers”) provides a short biography of Adolf Hurwitz (1859–1919) and his lesser-known older brother Julius Hurwitz (1857–1919). Both shared a passion for mathematics since their youth, but due to family circumstances, only Adolf could enter a university, while Julius became a banker. Nevertheless, he returned to school at the age of 33, graduated from a Realgymnasium and joined his brother Adolf, who held a professorship at the University of Königsberg. After Adolf accepted a position at the Zürich Polytechnic, Julius enrolled at the University of Halle-Wittenberg, where he successfully defended his doctoral thesis dealing with continued fraction expansion of complex numbers. He subsequently obtained a position at the University of Basel, but suffered from poor health, and his academic career was a short one. More information on Julius Hurwitz can be found in the author’s paper [Math. Intell. 39, No. 1, 44–49 (2017; Zbl 1436.01042)].
Part 2 (“Adolf Hurwitz’s mathematical diaries: an example of understanding mathematics development on the basis of historical documents”) contains more details on the topic of Julius Hurwitz’s doctoral thesis. Although his official advisor was Albert Wangerin, the topic was proposed by Julius’ brother Adolf, and was inspired by his earlier research. The mathematical notebooks of Adolf Hurwitz, which are now stored at the ETH Zürich, confirm that the two brothers exchanged ideas on complex continued fractions during the period when Julius was working on his thesis. More details about the development of complex continued fractions and the work of the Hurwitz brothers can be found in [the author and J. J. Steuding, Arch. Hist. Exact Sci. 68, No. 4, 499–528 (2014; Zbl 1304.01026)].
Part 3 (“A fruitful friendship: Adolf Hurwitz and David Hilbert”) makes use of Adolf Hurwitz’s notebooks to trace the lifelong friendship between Adolf Hurwitz and his student David Hilbert. It shows how Hilbert gradually emancipated from his teacher, but the two mathematicians maintained close contacts, which served as a fruitful source of new ideas for both of them.
Part 4 (“Julius Hurwitz and an ergodic theoretical view of his complex continued fraction”) is purely mathematical. It provides a link between Julius Hurwitz’s results on complex continued fractions and much more recent results obtained by S. Tanaka [Tokyo J. Math. 8, 191–214 (1985; Zbl 0581.10028)]. It turns out that Tanaka essentially rediscovered the same continued fraction expansion as Hurwitz without being aware of the latter’s dissertation thesis. This part also contains the author’s proof of an analogue of the Doeblin-Lenstra conjecture for the complex case of Tanaka’s continued fraction algorithm; cf. the author’s paper [“The Doeblin-Lenstra conjecture for a complex continued fraction algorithm”, Tokyo J. Math. 40, No. 1, 45–52 (2017; doi:10.3836/tjm/1502179214)].
For the entire collection see [Zbl 1364.11007].
Part 1 (“Meeting the Hurwitz brothers”) provides a short biography of Adolf Hurwitz (1859–1919) and his lesser-known older brother Julius Hurwitz (1857–1919). Both shared a passion for mathematics since their youth, but due to family circumstances, only Adolf could enter a university, while Julius became a banker. Nevertheless, he returned to school at the age of 33, graduated from a Realgymnasium and joined his brother Adolf, who held a professorship at the University of Königsberg. After Adolf accepted a position at the Zürich Polytechnic, Julius enrolled at the University of Halle-Wittenberg, where he successfully defended his doctoral thesis dealing with continued fraction expansion of complex numbers. He subsequently obtained a position at the University of Basel, but suffered from poor health, and his academic career was a short one. More information on Julius Hurwitz can be found in the author’s paper [Math. Intell. 39, No. 1, 44–49 (2017; Zbl 1436.01042)].
Part 2 (“Adolf Hurwitz’s mathematical diaries: an example of understanding mathematics development on the basis of historical documents”) contains more details on the topic of Julius Hurwitz’s doctoral thesis. Although his official advisor was Albert Wangerin, the topic was proposed by Julius’ brother Adolf, and was inspired by his earlier research. The mathematical notebooks of Adolf Hurwitz, which are now stored at the ETH Zürich, confirm that the two brothers exchanged ideas on complex continued fractions during the period when Julius was working on his thesis. More details about the development of complex continued fractions and the work of the Hurwitz brothers can be found in [the author and J. J. Steuding, Arch. Hist. Exact Sci. 68, No. 4, 499–528 (2014; Zbl 1304.01026)].
Part 3 (“A fruitful friendship: Adolf Hurwitz and David Hilbert”) makes use of Adolf Hurwitz’s notebooks to trace the lifelong friendship between Adolf Hurwitz and his student David Hilbert. It shows how Hilbert gradually emancipated from his teacher, but the two mathematicians maintained close contacts, which served as a fruitful source of new ideas for both of them.
Part 4 (“Julius Hurwitz and an ergodic theoretical view of his complex continued fraction”) is purely mathematical. It provides a link between Julius Hurwitz’s results on complex continued fractions and much more recent results obtained by S. Tanaka [Tokyo J. Math. 8, 191–214 (1985; Zbl 0581.10028)]. It turns out that Tanaka essentially rediscovered the same continued fraction expansion as Hurwitz without being aware of the latter’s dissertation thesis. This part also contains the author’s proof of an analogue of the Doeblin-Lenstra conjecture for the complex case of Tanaka’s continued fraction algorithm; cf. the author’s paper [“The Doeblin-Lenstra conjecture for a complex continued fraction algorithm”, Tokyo J. Math. 40, No. 1, 45–52 (2017; doi:10.3836/tjm/1502179214)].
For the entire collection see [Zbl 1364.11007].
Reviewer: Antonín Slavík (Praha)
MSC:
| 01A55 | History of mathematics in the 19th century |
| 01A60 | History of mathematics in the 20th century |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
| 11J70 | Continued fractions and generalizations |
| 11-03 | History of number theory |
| 37A05 | Dynamical aspects of measure-preserving transformations |
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