Legend has it that at the age of three, when is mother forgot her umbrella in a carriage, he replied, "It was number 354," and the umbrella was quickly re-acquired.Landau attended the French Lycée in Berlin, graduating at the age of 16 which is two years earlier than was normal. He then studied mathematics at the University of Berlin. His doctoral work there was supervised by Frobenius, and Landau received his doctorate in 1899 for a thesis on number theory. Landau was always interested in mathematical puzzles and even before he received his doctorate he had published two books on mathematical problems in chess. On 9 June 1900 he wrote a letter from Paris, where he was studying, to Hilbert giving an outline of his ideas for proving the prime ideal theorem for algebraic number fields.
He submitted this habilitation thesis in 1901, only two years after his doctorate, consisted of his work on Dirichlet series, a topic in analytic number theory. Frobenius was somewhat critical of the area that Landau worked in, and remarked at times that Landau's work would cease to become important if the Riemann hypothesis were proved. There is little doubt that Frobenius was quite wrong in his assessment of Landau's mathematical talents, but this did not affect Landau's self-confidence in any way.
Landau taught at the University of Berlin as a privatdozent from 1899 until 1909. During this period his publication list grew rapidly, so that by 1904 his publications exceeded his age of 27 and by 1909 he had nearly 70 papers in print. While at Berlin his ability to teach became clearly evident. He taught courses for beginners, which he did not have to do, and also lectured on his own speciality of number theory. In addition he gave lecture courses on the foundations of mathematics, irrational numbers, and set theory. Schappacher notes in  however:-
... it should also be said that he tended not to have cordial relationships with his students, being rather an aloof person.In 1909 he was appointed to an ordinary professorship at Göttingen as successor to Minkowski. He had Hilbert and Klein as colleagues at Göttingen until Klein retired in 1913. The successor to Klein was not easily found. Carathéodory came and went in less than three years while he was followed by Hecke who also left quickly in 1919. Landau worked hard to have Schur fill the chair but, against Landau's wishes, Courant was appointed. However, despite his outstanding talents as both a teacher and researcher, Landau managed to annoy many of his colleagues at Göttingen with his somewhat arrogant manner. Segal writes in :-
Landau was also something of a cynical snob. The story is well known that he used to tell people who would ask for his address in Göttingen, "You'll find it easily; it's the most splendid house in the city."We give some examples of how he annoyed his colleagues. Usually it was because he privately, and often publicly, criticised their results, although one would have to say that Landau was extremely knowledgeable and was almost always mathematically correct :-
Landau was a mathematician of encyclopaedic knowledge of the literature in his special areas of expertise, meticulous to a fault, and always devoted to finding the simplest possible result.After Koebe and Bieberbach had disputed in 1921 the significance of certain results each had published, Landau entered the argument the following year by writing a joint letter to the two of them in which he said that Koebe was the more correct of the two, but still not correct enough. He also published simplified proofs of some theorems of Bieberbach, and gave stronger results. Of course usually mathematicians are delighted to see others using their results to push forward, and consider it a compliment if someone publishes a new proof of their results, but it seems to be the arrogant way that Landau did such things that annoyed his colleagues. Landau also criticised proofs of theorems published by Blaschke, again saying that the proofs were unnecessarily complicated. One he claimed was trivial and another could be trivially deduced from a theorem of Mittag-Leffler. A letter which Blaschke sent to Bieberbach in 1921 ends with the sentence:-
Wouldn't you like to free Göttingen from Landau?Despite having friends who were well informed as to what the Nazis might do if they came to power, Landau failed to recognise the danger. In 1932 he was visited by a friend Fritz Rathenau who told him that if the Nazis gained control they would build concentration camps in which to put Jews. Landau is said to have replied :-
In that case I should immediately reserve for myself a room with a balcony and a southern exposure.On 30 January 1933 the National Socialist party led by Hitler did come to power in Germany and Fritz Rathenau's predictions soon came true. The Civil Service Law was passed on 7 April 1933 which provided the means of removing Jewish teachers from the universities. In fact before any official work reached Göttingen from the Ministry, the Dean wrote to Landau on 28 April asking him not to give his summer lecture courses and these were given instead by Landau's assistant. Having received no further advice from the university authorities, Landau decided to give his autumn lectures as advertised. Schappacher, in , quotes a letter from Landau in which he described in unemotional terms what happened on the first day of lectures:-
On 2 November, about 11.15, as I wished to leave my office and go to the large lecture theatre to begin my lecture, the entrance hall was filled with about 80 to 100 students who let me pass through unhindered. In the lecture hall was one person. Clearly therefore, there was a boycott with sentries at the door who had prevented (without force) those students who wanted to work from setting foot in the lecture room.Teichmüller, as leader of the students, had organised the boycott of Landau's lectures. In fact Teichmüller went to Landau's office after the boycott and explained that it was not the work of any organised group. Landau then asked Teichmüller to put this in writing and he included it with his request to the Ministry that he be retired. Despite Teichmüller's assertions, it is believed that student members of the Sturmabteilung (Stormtroopers) organised the boycott. Landau was given permission on 19 November to work at Groningen, in the Netherlands, and the permission was later extended to allow him to remain there for the winter semester. He was officially retired on 7 February 1934, moved to Berlin and after this only lectured outside Germany, spending some time in Cambridge and in Holland. He received full pay until 1 July 1934, then a pension until his death from a heart attack. His widow continued to receive a pension but in March 1939 she was informed that her pension would be terminated if she emigrated to the United States.
What happened - and it happened with the collaboration of many who were my pupils - leads me to believe that the only consequence must be my application to become emeritus or pensioned.
Landau's main work was in analytic number theory and the distribution of primes. He gave a proof of the prime number theorem in 1903 which was considerably simpler that the ones given in 1896 by Vallée Poussin and Hadamard. One consequence of his simpler proof was that it enabled him to obtain results concerning the distribution of prime ideals in algebraic number fields. His masterpiece of 1909 was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory. He also wrote important works on the theory of analytic functions of a single variable. Darstellung und Begründung einiger meuerer Ergebnisse der Funktiontheorie :-
... contains a collection of interesting and elegant theorems of the theory of analytic functions of a single variable. Landau himself discovered some of the theorems and demonstrated others in a new and simpler fashion.Schoeneberg writes :-
Written with the greatest care, Landau's books are characterised by argumentation which is complete, and as simple as possible. The necessary prerequisite knowledge is provided, and the reader is led securely, step by step, to the goal.Landau wrote over 250 papers on number theory which had a major influence on the development of the subject.
Article by: J J O'Connor and E F Robertson