Landau on Number Theory 401 Landau on Number Theory

Vorlesungen iiber Zahlentheorie. By Edmund Landau. Volume I: Aus der elementaren unà additiven Zahlentheorie. xii+360 pp. Volume II: Aus der analytischen und geometrischen Zahlentheorie. vii+308 pp. Volume III : Aus der algebraischen Zahlentheorie und iiber die Fermatsche Vermutung. vii+341 pp. Leipzig, S. Hirzel, 1927. These three excellently printed and arranged volumes form an addition of the highest importance to the literature of the theory of numbers. With them, the reader familiar with the basic elements of the theory of functions of a real and complex variable, can follow many of the astonishing recent advances in this fascinating field. His interest is enlisted at once and sustained by the accuracy, skill, and enthusiasm with which Landau marshals the analytic facts and simplifies as far as possible the inevitable mass of details. Part I of Volume I gives a treatment of the elements of the theory of numbers up to the theory of quadratic residues and of the Pellian equation. This occupies a little over 50 pages. In Part II is first deduced the elementary inequality