Edmund Landau

"Any book that revisits the foundations of analysis has to reckon with the formidable precedent of Edmund Landau's Grundlagen der Analysis (Foundations of Analysis) of 1930. Indeed, the influence of Landau's book is probably the reason that so few books since 1930 have even been attempted to include the construction of the real numbers in an introduction to analysis. On the other hand, Landau's account is virtually the last word in rigor. The only way to be more rigorous would be to rewrite Landau's proofs in computer-checkable form—which has in fact been done recently. On the other hand Landau's book is almost pathologically reader-unfriendly. ...While memories of Landau still linger, so too does fear of the real numbers. In my opinion, the problem with Landau's book is not so much the rigor (though it is excessive), but the lack of background, history, examples, and explanatory remarks. Also, the fact that he does nothing with the real numbers except construct them. In short, it could be an entirely different story if it were explained that the real numbers are interesting! That is what I have tried to do...."

"It is one thing to reflect, as Bieberbach did, for instance on the relative pedagogical merit of different ways to introduce π in a calculus class: geometrically via the circle, or in Landau's way via the zeroes of the cosine, with this function being defined by the power series. And it is quite a different matter to use such reflections as a basis for the forced removal of a distinguished colleague from teaching. Bieberbach's behaviour came all the more as a shock as nothing in his previous biography seemed to prepare one for it..."

"Landau was the son of a well-to-do Berlin gynecologist (who invented the myomectomy operation)... his mother was from the banking family of Jacoby, and Landau grew up in a Jacoby house amid other Berlin banking families. ...Landau married Marrianne Erlich, daughter of Paul Ehrlich... Ehrlich had been a fellow student with Landau's father. Thus Landau grew up a well-connected and well-to-do person... he was also something of a prodigy. Legend has it that at age three, when his mother forgot her umbrella in a carriage, he replied, "It was number 354," and the umbrella was quickly reacquired. ...Landau was also something of a cynical snob."

"Wir Mathematiker sind alle ein bißchen meschugge."

"Please don't read the preface for the teacher."

"I will ask of you only the ability to read English and to think logically—no high school mathematics, and certainly no higher mathematics."

"The multiplication table will not occur in this book, not even the theorem,2 \cdot 2 = 4,but I would recommend, as an exercise, that you define2 = 1 + 1, 4 = (((1 + 1) + 1) + 1)and then prove the theorem."

"My book is written, as befits such easy material, in merciless telegram style ("Axiom," "Definition," "Theorem," "Proof," occasionally "Preliminary Remark")... I hope I have written this book in such a way that a normal student can read it in two days. And then (since he already knows the formal rules from school) he may forget its contents."

"In 1933 Landau was dismissed from his [University of Göttingen] chair on the grounds of his race. An important colleague... Ludwig Bieberbach ...wrote the following lines in a treatise on Personality structure and mathematical creativity: "In this way... the ultimate reason behind the courageous rejection which the students at Göttingen University meted out to a great mathematician, Edmund Landau, was that his un-German style in research and teaching had become intolerable to German sensitivities. A people which has seen how alien desires for dominion are gnawing at its identity, how enemies of the people are working to impose their alien ways on it, must reject teachers of a type alien to it." The English mathematician Godfrey H. Hardy... responded to Bierbach... "There are many of us, many English and many Germans, who said things during the (First) War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all costs not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanation for his utterances; and I find myself driven to the more uncharitable conclusion that he really believes them true.""

"He possessed an enormous capacity for work - up to 12 or more hours a day. ... He worked to completely rigorous rules. We were once 'working' together in Cambridge, and started immediately after breakfast. I presently said: excuse me for a minute or two. 'Two minutes 47 seconds.' ... He was completely non-musical (as were Klein and Hardy). ... When G. H. Hardy wrote after the First World War to the effect that he had not been a fanatical anti-German, and felt confident that Landau would wish to resume former relations, Landau replied: 'As a matter of fact my opinions were much the same as yours, with trivial changes of sign.'"

"Not only the physical but also the intellectual landscape of German-language mathematics in the early 1930s would be impossible to imagine without German-Jewish mathematicians. Indeed, some fields of mathematics were completely transformed by their contributions. Number theory was transformed by Hermann Minkowski and Edmund Landau, algebra by Ernst Steinitz and Emmy Noether, set theory and general topology by Felix Hausdorff, Abraham Fraenkel and several others—to mention but a few examples."

"The principal events... took place in the early months of 1933... By April the Nazis had almost total control of Germany. One of their first decrees, on April 7, was intended to bring about the dismissal of all Jews from the civil service. ...University professors were civil servants ...Of the five professors teaching mathematics at Götingen, three—Edmund Landau, Richard Courant, and Felix Bernstein—were Jewish. A fourth, Hermann Weyl, had a Jewish wife. ...the April decree did not apply to Landau or Courant, since they fell within the Hindenburg exceptions. ...It did not help that Götingen at large was rather strong for Hitler. This was true of both "town" and "gown." ...(That grand house of which Edmund Landau was so proud had been defaced with a painting of the gallows in 1931.) On April 26 the town newspaper... printed an announcement that six professors were being placed on indefinite leave. ...One holdout was Edmund Landau (the only Götingen math professor... who was a member of the town's synagogue). Relying on the integrity of the law, Landau attempted to resume calculus classes in November... but the Science Student's Council... organized a boycott. Uniformed storm troopers prevented Landau's students from entering the lecture hall. With singular courage, Landau asked the Council leader, a 20-year-old student named Oswald Teichmüller, to write out as a letter his reasons... his reasons were ideological. He... felt it improper that German students should be taught by Jews. We are accustomed to think of Nazis activists as thugs, low-lifes, opportunists and failed-artists... which, indeed, most of them were. ...they also included in their ranks some people of the highest intelligence."

"Hilbert's solution of Waring's Problem was ready to be presented in the joint seminar with Minkowski in the middle of January 1909. After Minkowski's death, Hilbert presented his solution to Göttingen Academy on 6 February 1909, dedicating it to the memory of his friend, who had done so much for number theory. From then on he missed Minkowski, but carried on the seminar with Edmund Landau, Minkowski's successor. In looking for a successor to Minkowski, Klein and Hilbert looked for a young mathematician, whose achievements were still ahead of him. This requirement ruled out Adolf Hurwitz, and the final candidates were Oskar Perron and Edmund Landau. The decision was made by Klein, who said: 'Oh, Perron is such a wonderful person. Everybody loves him. Landau is very disagreeable, very difficult to get along with. But we, being such a group as we are, it is better that we have a man who is not easy'. Landau, though a worthy successor with respect to number theory... showed no interest in geometry and even less in applied mathematics, not to speak of mathematical physics. ...Hilbert knew that in executing his plans concerning physics, he could not count on Landau."

"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics. A striking example is that of the distribution of prime numbers. The solution of this problem lies in finding a general formula which tells us the number of primes that lie in any given numerical interval. ...Edmond Landau ...wrote two large volumes analyzing this problem without solving it, using the most advanced mathematics known at the time. Even in the elementary aspects of mathematics we are thus dealing with complex topics which make great demands on our mathematical skills."

"The present work is inspired by Edmund Landau's famous book, Handbuch der Lehre von der Verteilung der Primzahlen, where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers"."

"Die Zahlentheorie ist nützlich, weil man mit ihr promovieren kann."

"Bitte vergiß alles, was Du auf der Schule gelernt hast; denn Du hast es nicht gelernt."