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Collected works. Ed. by P. T. Bateman, L. Mirsky, H. L. Montgomery, W. Schaal, I. J. Schoenberg, W. Schwarz, H. Wefelscheid. With the assistance of P. L. Butzer, W. Heise, D. Laugwitz, A. Pfister, E. L. Stark. Volume 3. (English) Zbl 0653.01020

Essen (FRG): Thales Verlag. 544 p. DM 224.00 (1986).
See reviews of Volumes 1 and 2 in Zbl 0653.01018) and Zbl 0653.01019 above.
This volume opens with detailed commentaries on many of the papers in volumes 1 and 2, bringing the matters treated in them up to the present state of knowledge. In fact all but 8 of the 43 papers in these two volumes receive some sort of comment. In addition there is commentary on paper # 55 (in this volume) which contains Landau’s proof of the Weierstrass approximation theorem; the first proof in which “the approximating functions are ‘visibly’ polynomials” (to quote Hardy and Heilbronn), as well as on paper # 141 (on sums of squares in an algebraic number field) in a much later volume. These range from brief remarks to short surveys (e.g. on Landau # 2, the eight-queens problem – the queens problem on an \(n\times n\) board is still unsolved) to a 30- page expository survey by Butzer and Stark on Landau’s singular integral and Landau polynomials (Landau # 55).
The papers printed in this volume are numbered 44 to 56 and include in addition the already mentioned paper # 55, a long survey with Landau’s customary clarity of the circle of ideas connected with Picard’s theorem as they had developed by 1906 (# 44), and a long memoir on the multiplication problem for Dirichlet series (# 54) which would later find place in Landau’s prime number theory book. Both of these are in Landau’s expansive pre-World War I style. A 134-page memoir of contributions to analytic number theory (1908) is the last paper in the volume.
Unfortunately, the commentary in the volume, both editorial and mathematical, is marred by inelegancies of phrasing and poor proofreading. This seems particularly ironic in an edition of Landau’s collected works; nevertheless there are no mathematically disturbing errors, and this comment in no way detracts from the debt of gratitude owed the editors and commentators for making this edition possible.
The frontispiece of this volume is a picture taken in 1931 at MGM in Hollywood of Edmund and Marianne Landau with their daughter Susanne, Gordon Whyburn, Earl Hedrick, and Karl Menger.

MSC:

01A75 Collected or selected works; reprintings or translations of classics
11-03 History of number theory
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