Died: 25 February 1988 in Canberra, Australia

**Kurt Mahler**'s parents were Hermann Mahler (1858-1941) and Henriette Stern (1860-1942). The Mahler family were Jewish and long established in the Prussian Rhineland. Hermann Mahler had become an apprentice bookbinder, working his way up to become the owner of a small printing and bookbinding firm. In this respect he was following the Mahler family tradition of working in the printing and bookbinding trade. Hermann and Henriette Mahler had eight children with Kurt and his twin sister Hilde being the youngest. Of the six older children, Lydia married a printer, Josef took over Hermann's printing firm but died in a Nazi concentration camp during World War II, while the other four children died young. Mahler recalled that when he was a child, their home was [1]:-

... run on strictly orthodox Jewish lines and we were also good German patriots.

Although there was no academic tradition in the family, the [1]:-

... children acquired a love of reading from their father. "When I was not playing as a child with some mechanical toy", he recalled "then I was certainly reading". In particular, there was an elementary book on geometry which he would not then understand, but of which he liked to copy the figures.

Kurt contracted tuberculosis at the age of five years and, as a result, had severe problems with his right knee. He underwent several operations on this knee over the following years. Because of these health problems he attended school for only four years leaving at Easter 1917 at the age of 13. His parents arranged some private tuition for him at home, but he also attended some technical schools to learn to become an instrument maker. Of course, this led to him being introduced to a certain amount of mathematics which he loved [3]:-

He very quickly decided that mathematics was what he really liked doing. Already, from the summer vacation of1917, he began teaching himself logarithms(the arithmetic properties of which turned out to be one of his abiding interests in transcendental number theory)plane and spherical trigonometry, analytic geometry and calculus.

In 1918, at the age of fifteen, he took a job as an apprentice in a machine factory in Krefeld. He worked there for almost three years, the first of which he spent in the drawing office, the rest of the time being spent working in the factory itself. He was self-taught in mathematics teaching himself while working in the factory. He read works by Edmund Landau, Konrad Knopp, Felix Klein and David Hilbert among others, a remarkable achievement for someone with little mathematical background and nobody to offer guidance. His father sent small articles his son had written to Josef Junker, the head of the local high school. Junker, who had a doctorate in invariant theory written with Elwin Christoffel as his advisor, was impressed by Mahler's articles and sent them to Felix Klein who, in turn, passed them to his assistant Carl Siegel. Siegel suggested that Mahler should attend University. However, before entering university, Mahler had to gain the necessary entrance qualifications. He left his apprenticeship at the machine factory and worked at home preparing to take the Abitur examination. He received some help from the local high school teachers for the papers on German, French, and English which he was required to take, continuing to study mathematics on his own. He passed the Abitur in 1923 but he felt that he "just scraped through". The article [10] is an essay *Why I have a special liking for mathematics* written in 1923. It:-

... is an enthusiastic and moving confession of the author's interest in mathematics. It is followed by a brief account of his(unusual and remarkable)educational experience up to the beginning of his university studies. The essay was written to demonstrate to Mahler's German teacher(at the Realschule in Krefeld)that "there was at least one subject in which I was truly interested".

By the time that Mahler had passed his Abitur, Carl Siegel had moved to the University of Frankfurt and he arranged for him to study there. At Frankfurt, supported financially by his parents and several members of the Krefeld Jewish community, he attended lectures by Max Dehn on topology, Ernst Hellinger on elliptic functions, Carl Siegel on calculus and Otto Szász [2]:-

Mahler was clearly greatly influenced during this period by Siegel, who was the only person whom he recognized as his teacher in mathematical research.

In 1925 Siegel left Frankfurt for a period of overseas visits, and Mahler moved to Göttingen where he attended lectures by Emmy Noether, Richard Courant, Edmund Landau, Max Born, Werner Heisenberg, David Hilbert and Alexander Ostrowski, and acted as an unpaid assistant to Norbert Wiener. It was through lectures by Emmy Noether that he learnt about *p*-adic numbers which were to be one of the major topics of his research throughout his life. In 1927 he submitted his doctoral dissertation *Über die Nullstellen der unvollständigen Gammafunktion,* [on zeros of the incomplete gamma function], to the University of Frankfurt. He remained at Göttingen where he was supported by a research fellowship from the Notgemeinschaft der Deutschen Wissenschaft. During his tenure of the fellowship [1]:-

... he developed a new method in transcendental theory, found his celebrated classification of transcendental numbers and pioneered diophantine approximation in p-adic fields.

Papers he published during this period include: *Über einen Satz von Mellin* (1928); *Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen* (1929); *Über die Nullstellen der Abschnitte der hypergeometrischen Reihe* (1929); *Zur Fortsetzbarkeit gewisser Dirichletscher Reihen* (1930); *Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen* (1930); *Ein Beweis des Thue-Siegelschen Satzes über die Approximation algebraischer Zahlen für binomische Gleichungen* (1931), *Über das Mass der Menge aller S-Zahlen* (1932); and *Zur Approximation algebraischer Zahlen* (1933).

In 1933 Mahler was appointed to his first post at the University of Königsberg but, before he could take up the post, Hitler came to power. Mahler realised at once that, as he was Jewish, he had to leave Germany. After visiting van der Corput, and his two pupils Koksma and Popken, in Amsterdam during the summer of 1933, he accepted an invitation from Louis Mordell to go to Manchester where he spent 1933-34 supported by a Bishop Harvey Goodwin Fellowship. He spent 1934-36 in Groningen in the Netherlands supported by a fellowship from a Dutch Jewish group which van der Corput had arranged. He attended the International Congresses of Mathematicians at Oslo in July 1936 where he met Paul Erdős for the first time. Erdős writes [4]:-

I knew of his work many years earlier and was very glad to meet him. I almost immediately posed him the following problem: An integer is called powerful if p|m implies p^{2}|m; are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes!x^{2}- 8y^{2}= 1has infinitely many solutions. I was a bit crestfallen since I felt that I should have thought of this myself.

However, back in Groningen, Mahler was involved in a bicycle accident and his knee troubles returned. He underwent several operations on his knee back home in Krefeld, where eventually his kneecap was removed, and also spent some time in Switzerland during the summers of 1937 and 1938 where he was finally cured. However, despite being cured, he walked with a limp for the rest of his life. Mahler returned to Manchester in 1937, where again he interacted with Erdős [4]:-

We wrote two joint papers, had many mathematical and political discussions, walked a great deal(despite his poor health, Mahler liked to walk very much)and we also played bridge.

However, during 1940 he was interned as "an enemy alien" for three months and spent some time in the same camp on the Isle of Man as Kurt Hirsch. These three years had been highly productive on the mathematical side, but very difficult financially for, despite two short-term temporary assistant lectureships and some funds from a fellowship, he mainly had to support himself from his savings. He had learnt Chinese in 1939 in preparation for getting a post in China, but the onset of the war prevented this being realised. Returning to Manchester after his period of internment, he was appointed as an assistant lecturer in 1941 (his first permanent post) and remained there until 1962. His outstanding mathematical achievements led to steady promotion: Lecturer in 1944, Senior Lecturer in 1947, Reader in 1949, and a personal professorship in 1952. It is worth noting that his personal professorship was the first in Manchester, and was specially created to acknowledge his eminent status. Paul Cohn arrived in Manchester as an assistant lecturer in 1952 and he, like Mahler, lived in Donner House. Cohn writes [1]:-

The only other mathematician I found there was Professor Mahler, so we saw a good deal of each other for the next six years. He was without any pretensions and one could discuss anything under the sun with him, though preferably mathematics, photography or Chinese(in that order). For his holidays he always went to the island of Herm, saying that it suited him because he could not walk far. The island could be reached only by a small boat from Guernsey; it had no motor traffic and could be crossed in half an hour walking slowly. ... His attitude to mathematics was like his attitude to life: he liked things as simple as possible and usually eschewed abstraction, but with his direct methods was often able to go surprisingly far. ... He was generally very precise and punctual, rising and going to bed early. At his request seminars at Manchester were held at2p.m. If a speaker was still on his feet at3.01, Mahler(who always sat in the front row)would open and shut his little attaché-case repeatedly with a loud click.

In 1962 he went to Canberra for the last 6 years of his career. This was a purely research appointment but Mahler agreed to give an undergraduate number theory course. One of the undergraduate students who took this course was John Coates. After officially retiring from Canberra in 1968, Mahler accepted an invitation from Ohio State University where he worked until 1972 when he returned to Canberra [1]:-

As he grew older, deteriorating health made it more and more difficult for him to travel, but he remained mathematically active until the end.

Erdős writes [4]:-

I visited Australia fairly often and of course always visited Mahler in Canberra. We had many mathematical discussions but he could no longer walk a great deal. I was in Australia for over two months early in1988and in Canberra in February. I had dinner and lunch with Mahler at University House and met him in the Department of Mathematics of the Institute for Advanced Studies. He was clearly frail but I did not expect that the end was so near.

Mahler published around 200 papers. He worked on transcendence of numbers, showing in 1946 that

0.123456789101112131415161718192021...

was transcendental. In [13] it is noted that:-

... Mahler regretted that, apart from his own work, little interest had been shown by20th century mathematicians in the study of arithmetical properties of decimal expansions.

He also classified real and complex numbers into classes which are algebraically independent. Other major themes of his work were rational approximations of algebraic numbers, *p*-adic numbers, *p*-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials. He proved important results about polar convex bodies, compound convex bodies and the very useful Mahler Compactness Theorem. He published a number of excellent books, but these were all based on lecture courses he had given, often using notes taken by someone who attended the course. For example *Lectures on diophantine approximations : g-adic numbers and Roth's theorem* (1961) was prepared from notes by R P Bambah of lectures given by Mahler at the University of Notre Dame in autumn 1957 and was described as an "extremely valuable contribution". In the Preface to *Introduction to p-adic numbers and their functions* (1973) Mahler writes:-

This set of notes contains an elementary introduction to the theory of p-adic numbers and their analysis. These numbers were introduced by K Hensel some eighty years ago and have slowly become of importance in more and more parts of mathematics. Nevertheless, while many recent books on algebra have short chapters or paragraphs on the subject, a really good introduction to p-adic numbers from the standpoint of elementary analysis does not seem to exist. ... We shall begin by studying the g-adic rings and p-adic fields, and then finally investigate continuous and differentiable functions of a p-adic variable. A course similar to this presentation was given repeatedly at the Ohio State University.

A second, improved and expanded, edition of the book was published in 1981:-

The text is written in complete detail, sometimes several proofs of a result are given and lots of concrete examples and special cases are calculated(the results of which are also useful for the specialist).

Mahler's *Lectures on transcendental numbers* (1976) was based on lectures given twenty years earlier.

Mahler received many awards. He was elected a Fellow of the Royal Society in 1948. The London Mathematical Society awarded him its Senior Berwick Prize in 1950 and its De Morgan Medal in 1971. He was elected a fellow of the Australian Academy of Sciences in 1965 and awarded its Lyle Medal in 1977. He was elected an honorary member of the Dutch Mathematical Society in 1957 and the Australian Mathematical Society elected him an honorary member in 1986.

**Article by:** *J J O'Connor* and *E F Robertson*

**December 2008**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Mahler.html]