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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. Volume 8 (with comments by Jean Dieudonné, Hans Zassenhaus, W. Narkiewicz and Peter Bundschuh). (German) Zbl 0665.01016

Essen: Thales Verlag. 432 p. DM 224.00 (1987).
[For volume 7, see above.]
In addition to the reproduction of papers #176–204 (from the years 1923–1926) of Landau, this volume also contains some commentary on some of the papers. These are quite varied in nature and style. There are two – plus pages by Jean Dieudonné summarizing Landau’s work in complex variables. Hans Zassenhaus provides very brief comments on three papers (#89 in vol. 5, #118 in vol. 6 and the posthumous #253 in vol. 9), all of which essentially deal with lattice points. Much more informative than either of these is the substantial discussion by W. Narkiewicz of 11 of Landau’s papers in algebraic number theory. These are arranged as commentary on the individual papers, and discuss results and history, with the problems treated up until the present day, and consequently are quite useful. The papers treated span the years 1911- 1935. Three of these papers deal with Gauss’ class number conjecture, and together with a fourth on class numbers (#17 of 1903) are discussed in a short and informative essay by Peter Bundschuh. Inevitably there is considerable overlap between this essay and Narkiewicz’ contribution.
Also in this volume is an eight-page anecdotal and amusing essay in a rambling style by Laurence Young entitled “The place of Edmund Landau in XXth century mathematics”. While the preservation for the future of the content of this interesting essay is valuable, it does not really address its title.
As has been true throughout these volumes, the proofreading of the nonreproduced material prepared for this volume is very bad (including in one place writing 12 for 23). As mentioned earlier this is especially ironic in a set of volumes devoted to Landau’s collected works. The frontispiece is a formal “bust portrait” photograph of Landau as a young man. Again the source of the photograph is not indicated.
As to mathematical content, most of the papers in this volume deal with lattice-point problems. These include a joint paper with Hardy on the “Hardy-Landau identity” (#181) and the well-known “The unimportance of Pfeiffer’s method for analytic number theory” (#190). In the twelve years since Landau had “unearthed” and made rigorously usable “Pfeiffer’s method”, (#90 in vol. 5) a number of mathematicians including Landau himself had obtained significant results using it. Van der Corput had already somewhat simplified the method, and in this paper Landau shows how all the geometric “Pfeiffer-type” considerations can be replaced by simpler Fourier-series sort of arguments. Paper #199 is an exposition in Italian of this new method restricted to the lattice points of a circle. In 1925 Landau was 48 and he writes in #190 “I will lead the reader up to the threshold of the chaos [caused by Van der Corput’s reducing the exponent in the error term for the circle problem to \(<1/3]\), by a very comfortable path; youthful researchers have it easier than we older ones who must build the first routes.”
However, in addition to the contributions to lattice point problems, this volume shows a continuation of work on \(\zeta\) (s) (including the collaboration with Harald Bohr), and some papers on Waring’s Problem, as well as brief notes on other topics. Paper #198 on Waring’s Problem is a correct version of Vinogradov’s method; and while Landau mentions the numerous errors in Vinogradov’s original, he says “[Vinogradov’s] paper has plainly been printed before being carefully worked through. Nevertheless, the following cleaned-up version will show for how much science owes gratitude to Vinogradov.” Paper #195, a survey article on problems in elementary number theory, was delivered in Hebrew at the Inauguration of the Hebrew University in Jerusalem and is so reproduced (for Landau and Hebrew, see A. A. Fraenkel’s Lebenskreise). Paper #192 is a 49 page monograph on inequalities between derivatives written for the Royal Danish Academy.
The papers of this volume end in 1926. 1927 would see publication of the second edition of his “Algebraische Zahlen” and the monumental three- volume Vorlesungen über Zahlentheorie. Some of the papers in this volume probably arose through working out clear proofs for this latter book.
Reviewer: S.L.Segal

MSC:

01A75 Collected or selected works; reprintings or translations of classics
01A60 History of mathematics in the 20th century
11-03 History of number theory
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