For many years the Viennese number theorist Edmund Hlawka was a truly major player of the game, so it is entirely appropriate that a selecta of his work should appear. In point of fact, it appeared already around 1990, nearly two decades before Hlawka’s passing at over ninety years of age. It is noteworthy that one of the later articles in the Selecta (dater 1980) should be titled “90 Jahre Geometrie der Zahlen” (i.e., “90 years of the geometry of numbers”), seeing that Hlawka was a leading expert in this field, active during many of the ninety years he references. This article by itself is worth the price of admission, being clearly and elegantly written (Hlawka’s German is easy to read), and covering great mathematical developments and events with style. It naturally starts with Minkowski (two long sections and one short one), then Blichfeldt, then Siegel and Mahler. After that Hlawka turns to topological and analytic (measure theoretic) aspects of geometric number theory, all making for a wonderful overview by a master of the trade.
Naturally, the majority of the entries in the book are reprints of Hlawka’s research articles, starting already with a 1938 work on complex inhomogeneous linear forms. Thereafter we encounter all sorts of marvels, not just from the theory of numbers, but from such areas as analysis (“Discrepancy and Riemann integration,” from 1971), the kinetic theory of gases, numerical mathematics and numerical analysis in particular, and so forth. However, the most frequent theme is Gleichverteilung, i.e. equidistribution.
Except for three in English, the articles are in German. But it is clear and concise German, not the prose of, say, Karl Rahner. It goes without saying that the mathematics is elegant and interesting, and it would serve any one in the field of number theory (of whatever flavor) to read these masterly crafted presentations. Finally, the last article in the book is Hlawka’s 1985 J. Number Theory biographical article on the recently deceased Carl Ludwig Siegel: one classical master talking about another — fabulous.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.