Number Theory and Analysis

February 14, 1968 marked the thirtieth year since the death of Edmund Landau. The papers of this volume are dedicated by friends, students, and admirers to the memory of this outstanding scholar and teacher. To mention but one side of his original and varied scientific work, the results and effects of which cannot be dis­ cussed here, Edmund Landau performed one of his greatest services in developing the analytic theory of prime numbers from a subject accessible only with great difficulty even to the initiated few to the general estate of mathematicians. With the exception of the work of Chebyshev, Riemann, and Mertens, before Landau the problems of this theory were attempted only in a number of papers which were filled with gaps and errors. These problems were such that even Gauss abandoned them after several attempts in his youth, and they were described by N. H. Abel in a letter of 1823 and by O. Toeplitz in a lecture in 1930 as the deepest part of mathe­ matics. Clarification first began with the papers of Hadamard, de la Vallee Poussin, and von Mangoldt. At the end ofthe foreword to his work" Handbuch der Lehre von der Verteilung der Primzahlen" which appeared in 1909, Landau could thus remark with complete justification: " . . . The difficulty of the previously unsolved problems has frightened nearly everyone away from the theory of prime numbers.

Inhalt

On the large sieve method.- Über binäre additive Probleme gemischter Art.- How to extend a Calculus.- On the representation of positive integers as sums of three cubes of positive rational numbers.- Analytische Klassenzahlformeln.- Über Folgen ganzer Zahlen.- On the average length of a class of finite continued fractions.- Interpolation analytischer Funktionen auf dem Einheitskreis.- On the high-indices theorem for Borel summability.- Bemerkungen zu Landauschen Methoden in der Gitterpunktlehre.- Über einige Fragen der vergleichenden Primzahltheorie.- On local theorems for additive number-theoretic functions.- The "pits effect" for the integral function.- On numbers which can be expressed as a sum of powers.- On some Diophantine equations y2 = x3 + k with no rational solutions (II).- Über das Vorzeichen des Restgliedes im Primzahlsatz.- A measure for the differential-transcendence of the zeta-function of Riemann.- Comments on Euler's "De mirabilibus proprietatibus numerorum pentagonalium".- On the distribution of numbers prime to n.- Spline interpolation and the higher derivatives.- Zu den Beweisen des Vorbereitungssatzes von Weierstraß.- Über Gitterpunkte in mehrdimensionalen Kugeln IV.- Publications of Edmund Landau.