**Arnold Walfisz**'s father was Zelman** **Walfisz; the family were of Jewish origins. Arnold grew up in Warsaw where he attended the Sixth State Gymnasium and was awarded his Abitur in 1909. Later that same year, he went to Germany to begin his university education.

Perhaps we should look a little at the political situation at this time in order to understand the times in which Walfisz lived. Poland had ceased to become an independent country in 1795 when it was divided between Russia, Prussia and Austria. Warsaw was controlled by the Russians and, after various unsuccessful uprisings by Polish nationalists, a period of Russification followed. In particular, Warsaw University became a Russian university and much was spent on building a new campus. Walfisz preferred to go to Germany for his university studies. There he undertook the type of education which was typical of German students of this time, moving between various different German universities. In particular, he studied at the major German mathematical centres of Munich, Berlin, Heidelberg and Göttingen.

Walfisz was in the middle of his university studies when World War I began in 1914. Of course this war put Polish people in a somewhat difficult position since Poland was partitioned between countries which were on opposing sides. Most Poles sided with Central Powers which included Germany and Austria-Hungary, opposing the Allies mainly because they wished to be free of Russian domination. Walfisz, however, was not able to continue his education for the duration of the war, only being able to resume his studies in 1918. We have been unable to determine exactly what he did during the four years of the war. When he resumed his studies he undertook research for his doctorate, advised by Edmund Landau at the Georg-August University of Göttingen. He published his first paper *Über die Nichtfortsetzbarkeit einiger durch Dirichletsche Reihen definierter Funktionen*, written jointly with his thesis advisor Edmund Landau, in 1920. He submitted his 56-page thesis, *Über die summatorischen Funktionen einiger Dirichletsher Reihen* to Göttingen towards the end of 1921 and was awarded his doctorate in the following year.

Walfisz was not in a strong position to obtain a university post. As a Jew he was discriminated against within his own country of Poland, and as a Pole, he was discriminated against in Germany. We do not suggest that this made obtaining a post impossible, but it certainly made it much more difficult. Walfisz went to Wiesbaden after the award of his thesis and lived there for five years during which he produced many fine papers. He published four papers in 1924, three in 1925, one in 1926 and nine in 1927. All these papers were written in German, with the exception of *The lattice points of a circle* (a joint paper with J E Littlewood, written in English and published in 1924 by the Royal Society of London) and *New results in the theory of numbers* (Polish) published in 1925.

Walfisz had wanted to return to Göttingen and work with Edmund Landau. Indeed, Landau was also very keen to have Walfisz as an assistant so made application to the Rockefeller Foundation for financial support. However, despite the obviously strong academic case for such an award to be made, it was turned down by the Rockefeller Foundation. The reasons for turning down the application is discussed in [1] where two main reasons are suggested. First, the official reason was that Edmund Landau:-

... wished to engage him as a Private Assistant, in order to finish up an important piece of work on which he was engaged.

This reason was not in any way Walfisz's fault, yet it was he who would be the loser. A second reason for turning down the application was the following [1]:-

In the case of a candidate from "a somewhat scientifically backward country", there must be some indication that the home people will find a suitable place for him on his return from abroad. ... we have learnt from experience it is not profitable to take many men from the backward countries and help them to come to western Europe for what one might call general advanced training.

Of course, calling Poland at this time "a somewhat scientifically backward country" is rather ridiculous for mathematics flourished in Poland between the wars. However, it was certainly true that Walfisz would have had no "suitable place for him on his return from abroad".

In 1927 he returned to Warsaw where he took a job as a Mathematician with the Europa Insurance Company. He habilitated at the University of Warsaw in 1930 and taught there as a docent. However, he had to continue to earn a living with the insurance company since he was not paid for his university teaching. However, he lived most of the time in Radosc, a town halfway between Warsaw and Bialystok. Many of the papers that Walfisz published after returning to Poland give his affiliation as Warsaw but his address as Radosc. In 1935 Walfisz, together with Salomon Lubelski, founded the number theory journal *Acta Arithmetica*. He became a member of the editorial board and he published four papers in the journal around this time, two in 1935 ((with Sarvadaman Chowla) *Über eine Riemannsche Identität* and *Zur additiven Zahlentheorie*), and two in the following year (*Über Gitterpunkte in mehrdimensionalen Ellipsoiden V* and *Teilerprobleme V*). Let us note that Sarvadaman D S Chowla was an Indian mathematician who had studied for his doctorate at the University of Cambridge advised by John Edensor Littlewood. He obtained a doctorate in 1931 for the thesis *Contributions To The Analytic Theory Of Numbers*. In the same year Walfisz and Chowla had co-authored the paper *On a trigonometric sum* published by the London Mathematical Society.

The political situation in Poland was, by this time, rapidly deteriorating. Jozef Pilsudski, considered to be the founder of modern Poland, died in 1935 and a succession of military leaders ran the country. Border disputes with Germany and the USSR increased the instability and the Nazis looked ever more threatening to their neighbours and to all those of Jewish descent. Some Poles had already found positions in the USSR, such as Chaim (Hermann) Müntz, who had left Berlin to take up a chair for analysis at Leningrad in 1929. He influenced the Georgian University of Tbilisi to make Walfisz the offer of a position there. In October 1936 Walfisz accepted and began a career in Tbilisi. However, a look at the addresses on his letters he sent show that he visited Tbilisi before taking up the permanent appointment. For example, he corresponded with Louis Joel Mordell, who was at the University of Manchester, sending him several letters from Radosc from June 1934 to March 1935. However, he was in Tbilisi in January 1936, back in Radosc in August 1936, but writing from Tbilisi in October 1936. Similarly, he corresponded with Harold Davenport, all his letters between 1930 and 1936 being sent from Radosc.

Taking up permanent residence in Tbilisi in October 1936, Walfisz was appointed as a Senior Researcher at Tbilisi Mathematical Institute and as Professor of Mathematics at Tbilisi State University. From 1937 to 1940, he was Head of the Department of Number Theory at the Tbilisi Mathematical Institute and, from 1940 to 1944 he was Head of the Department of Theoretical Mathematics. He also served as Head of the Department of Algebra and Geometry at the Tbilisi Mathematical Institute from 1948 to 1962. For part of this time, from 1947 to 1953, he was also a Professor at the Tbilisi Pedagogical Institute. The authors of [6] write:-

Besides leading an intense scientific life Professor A Z Walfisz was active in many aspects of teaching. He gave lectures on different branches of mathematics at Tbilisi University and at the Tbilisi and Kutaisi Pedagogical Institutes. Moreover, he did a lot of work in training number theorists, a specialty that was not represented before in Georgia. His pupils now work prolifically in many educational establishments and Universities in the Republic of Georgia.

We have already seen something of his mathematics output, but this is summed up in [6] as follows:-

Walfisz published100scientific works in different branches of mathematics, mainly in number-theory. They were written in several languages and brought him world-wide recognition. The favourite field of research of Arnold Walfisz was the theory of integral points in ellipsoids, to which he devoted about thirty papers and a monograph.

He published *Gitterpunkte in Mehrdimensionalen Kugeln* in 1957. This German text was published in Warsaw by the Polish Academy of Sciences, but a Russian version of the text was published in Tbilisi by the Georgian Academy of Sciences. E M Wright, reviewing the German text, writes [10]:-

The author's style of exposition is as limpidly clear as Landau's and the book could be read with especial profit by anyone commencing work as a mathematical author. The beautiful theory is due to a sequence of celebrated mathematicians, including Landau, Hardy, Jarnik, Vinogradov, van der Corput and, most of all, Professor Walfisz himself. The book includes a detailed account of sources and a brief account of results in the more general problem for ellipsoids.

S Chowla, reviewing the same book, writes [3]:-

This is a beautifully written book by a leading expert in the field. Although of immense value to the specialist, it is addressed to a wider circle of readers.

Walfisz published further monographs: *Pell's equation *(Russian) (1952); *Lattice points in many-dimensional spheres* (Russian) (1960); and *Weylsche Exponentialsummen in der neueren Zahlentheorie* (1963). The first of these was in the Georgian Academy of Sciences' popular science series. B J Birch, reviewing the 1963 work, writes:-

This book contains a very detailed exposition of exponential sum estimates of N M Korobov and I M Vinogradov, and of the application of these estimates to obtain sharp results in the analytic theory of numbers.

We mentioned above that Walfisz co-founded the journal *Acta Arithmetica*, and served on the editorial board, when he was working in Warsaw. After he left Poland and went to Georgia in 1936 he could not continue with his editorial work on this journal and the journal ceased publication during World War II. However, after the Polish Academy of Sciences was founded in 1952, the Academy took over publication of the journal and invited Walfisz to join the editorial board. He served on the board for the rest of his life. Also, during the years he worked in Tbilisi, he served on the editorial board of the journal *Trudy Tbiliss koi matematicheskogo institut im. A M Razmadze Akademii Nauk Gruzinskoi SSR*. He also undertook other editorial work [6]:-

Walfisz also prepared the Russian translation of L E Dickson's "Introduction to the Theory of Numbers", which he considerably augmented and revised. Also worthy of mention is Edmund Landau's monograph "Diophantische Gleichungen mit endlich vielen Lösungen", which was published in1959by the German scientific press in Berlin(DDR)with a large supplement by Walfisz. The same press entrusted him, as a prominent pupil of Edmund Landau, with the publication of a collection of the papers of Landau on integral points. This appeared under the title "Ausgewählte Abhandlungen zur Gitterpunktlehre" with an extensive commentary by Walfisz.

We must not give the impression that all of Walfisz's mathematical contributions were in number theory for he also contributed papers on the theory of ideals in rings, on function theory, and the theory of modular forms.

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