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Hans Rademacher studied at Göttingen and was persuaded to study mathematics by Courant. His initial interests were in the theory of real functions which he was taught by Carathéodory who also taught him the calculus of variations. At Göttingen he also studied number theory with Landau. Continuing his interest in the theory of real functions he was awarded his doctorate in 1916 for a dissertation on single-valued mappings and mensurability.
He taught at a school in Thuringia for a short while before being appointed to the University of Berlin as a Privatdozent in December 1916. He was a colleague of Schmidt and Schur and was certainly influenced by them.
He changed his area of mathematical interest from the theory of real functions to number theory in 1922 when he accepted the position of extraordinary professor at Hamburg. He was led towards number theory by Hecke who had been appointed to Hamburg three years before Rademacher. At Easter 1925 Rademacher left Hamburg to become an ordinary professor at Breslau. It was a difficult decision for Rademacher, particularly since Hecke was so keen for him to stay in Hamburg. Had Hecke succeeded in his attempt to get Hamburg to offer Rademacher an ordinary professorship then he would almost certainly have remained there, but the university would not make the offer that Hecke requested and, after much thought, Rademacher went to Breslau.
In different political circumstances one would have expected Rademacher to remain at Breslau for the rest of his career. However when Hitler came to power in 1933 normal expectations were completely overturned for most people. Rademacher was not Jewish, and he was certainly racially acceptable to the Nazi regime, but his views were not acceptable to the Nazis. He was forced out of his professorship in 1933 because of his pacifist views and he left Germany in 1934. He spent the rest of his life in the United States, first at Swarthmore College, and later at the University of Pennsylvania.
Rademacher's early arithmetical work dealt with applications of Brun's sieve method and with the Goldbach problem in algebraic number fields. About 1928 he began research on the topics for which he is best known among mathematicians today, namely his work in connection with questions concerning modular forms and analytic number theory. Perhaps his most famous result, obtained in 1936 when he was in the United States, is his proof of the asymptotic formula for the growth of the partition function (the number of representations of a number as a sum of natural numbers). This answered questions of Leibniz and Euler and followed results obtained by Hardy and Ramanujan. Rademacher also wrote important papers on Dedekind sums and investigated many problems relating to algebraic number fields.
In addition to the significant contributions to real analysis and measure theory which we have briefly mentioned above, complex analysis, geometry, and numerical analysis. P T Bateman, reviewing Rademacher's collected works, wrote that they:-
... serve not only as a fitting memorial to a great mathematician and human being, but also provide excellent examples of how mathematics should be presented, and serve as leisurely but authentic introductions to some fascinating parts of analysis and number theory.
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