Hans Rademacher
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Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969,
Haverford, Pennsylvania Haverford is an Unincorporated area, unincorporated community located in both Haverford Township, Pennsylvania, Haverford Township in Delaware County, Pennsylvania, Delaware County, Pennsylvania, United States, and Lower Merion Township, Pennsylvan ...
, USA) was a
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...
-born American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, known for work in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
.


Biography

Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göttingen;
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
supervised his dissertation. In 1919, he became ''
privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...
'' under
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
at
University of Berlin The Humboldt University of Berlin (german: link=no, Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick Will ...
. In 1922, he became an assistant professor at the
University of Hamburg The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vor ...
, where he supervised budding mathematicians like Theodor Estermann. He was dismissed from his position at the
University of Breslau A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. ''University'' is derived from the Latin phrase ''universitas magistrorum et scholarium'', which ...
by the
Nazi Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right politics, far-right Totalitarianism, totalitarian political ideology and practices associated with Adolf Hit ...
s in 1933 due to his public support of the
Weimar Republic The German Reich, commonly referred to as the Weimar Republic,, was a historical period of Germany from 9 November 1918 to 23 March 1933, during which it was a constitutional federal republic for the first time in history; hence it is also r ...
, and emigrated from Europe in 1934. After leaving Germany, he moved to
Philadelphia Philadelphia, often called Philly, is the largest city in the Commonwealth of Pennsylvania, the sixth-largest city in the U.S., the second-largest city in both the Northeast megalopolis and Mid-Atlantic regions after New York City. Sinc ...
and worked at the
University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universit ...
until his retirement in 1962; he held the
Thomas A. Scott Professorship of Mathematics The Thomas A. Scott Professorship of Mathematics is an academic grant made to the University of Pennsylvania. It was established in 1881 by the railroad executive and financier: Thomas Alexander Scott. Recipients *Ezra Otis Kendall, 1881–1899 ...
at Pennsylvania from 1956 to 1962. Rademacher had a number of well-known students, including George Andrews,
Paul T. Bateman Paul Trevier Bateman (June 6, 1919 – December 26, 2012) was an American number theorist, known for formulating the Bateman–Horn conjecture on the density of prime number values generated by systems of polynomials and the Mersenne conjectures#Ne ...
, Theodor Estermann and
Emil Grosswald Emil Grosswald (December 15, 1912 – April 11, 1989) was a mathematician who worked primarily in number theory. Life and education Grosswald was born on December 15, 1912 in Bucharest, Romania. He received a master's degree in both mathemat ...
.


Research

Rademacher performed research in analytic number theory, mathematical
genetics Genetics is the study of genes, genetic variation, and heredity in organisms.Hartl D, Jones E (2005) It is an important branch in biology because heredity is vital to organisms' evolution. Gregor Mendel, a Moravian Augustinian friar worki ...
, the
theory of functions of a real variable In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
, and quantum theory. Most notably, he developed the theory of Dedekind sums. In 1937 Rademacher discovered an exact convergent series for the partition function P(n), the number of integer partitions of a number, improving upon Ramanujan's asymptotic non-convergent series and validating Ramanujan's supposition that an exact series representation existed.


Awards and honors

With his retirement from the
University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universit ...
, a group of mathematicians provided the seed funding for The Hans A. Rademacher Instructorships, and honored him with an honorary degree as Doctor of Science. Rademacher is the co-author (with
Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnas ...
) of the popular mathematics book ''The Enjoyment of Mathematics'', published in German in 1930 and still in print.


Works

* with
Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnas ...
: ''Von Zahlen und Figuren.'' 1930. 2nd edn. 1933. Springer 2001, . * ''
The Enjoyment of Mathematics ''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in ...
''. ''Von Zahlen und Figuren'' translated into English by Herbert Zuckerman, Princeton University Press, 1957 * with Ernst Steinitz ''Vorlesungen über die Theorie der Polyeder- unter Einschluss der Elemente der Topologie.'' Springer 1932, 1976. * ''Generalization of the Reciprocity Formula for Dedekind Sums.'' In: ''Duke Math. Journal.'' Vol. 21, 1954, pp. 391–397. * ''Lectures on analytic number theory.'' 1955. * ''Lectures on elementary number theory.'' Blaisdell, New York 1964, Krieger 1977. * with Grosswald: ''Dedekind sums.'' Carus Mathematical Monographs 1972. * ''Topics in analytic number theory.'' ed. Grosswald. Springer Verlag, 1973 (Grundlehren der mathematischen Wissenschaften). * ''Collected papers.'' 2 vols. ed. Grosswald. MIT press, 1974. *''Higher mathematics from an elementary point of view.'' Birkhäuser 1983.


Further reading

* George E. Andrews, David M. Bressoud, L. Alayne Parson (eds.) ''The Rademacher legacy to mathematics.'' American Mathematical Society, 1994. * ''Lexikon bedeutender Mathematiker.'' Deutsch, Thun, Frankfurt am Main, . * Tom Apostol: ''Introduction to Analytical number theory.'' Springer * Tom Apostol: ''Modular functions and Dirichlet Series in Number Theory.'' Springer * Obituary and list of publications.


See also

*
Hadamard transform The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal ...
* Rademacher's contour * Rademacher complexity * Rademacher function * Rademacher–Menchov theorem *
Rademacher's series In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function ...
*
Rademacher system Rademacher is an occupational surname of German origin, which means "wheelmaker". It may refer to: People *Arthur Rademacher (1889–1981), Australian football player * Autumn Rademacher (born 1975), American basketball coach * Bill Rademacher (bor ...
* Rademacher distribution * Rademacher's theorem


References


External links

* * 1892 births 1969 deaths 20th-century American mathematicians Mathematics popularizers German emigrants to the United States 20th-century German mathematicians Number theorists Mathematical analysts University of Pennsylvania faculty Mathematicians at the University of Pennsylvania New York University faculty University of Göttingen alumni University of Breslau faculty People from Wandsbek People from Hamburg University of Hamburg faculty Humboldt University of Berlin faculty {{Germany-mathematician-stub