**Paul Turán** was a Hungarian mathematician who worked in number theory.

- Paul was a brilliant pupil at secondary school in Budapest, showing at this stage his remarkable mathematical abilities.
- Turán entered Pázmány Péter University of Budapest already showing his potential for research.
- In 1933 Turán was awarded his diploma which qualified him to teach mathematics and science, and he continued working for his doctorate.
- It was significant in being Turán's first joint work with Erdős.
- It was not the result which Turán proved here that was significant, for he proved a result which had been known since 1917, namely that almost all integers nnn have asymptotically log log nnn prime factors.
- His Ph.D. was supervised by Fejér, and Turán was awarded the degree in 1935.
- Not only was 1938 significant in that Turán now at least had employment, but it was also the year in which he had his most fruitful mathematical idea.
- Turán mentioned these problems and told me that they were not only interesting in themselves but their positive solution would have many applications.
- Turán invented the power sum method while investigating the zeta function and he first used the method to prove results about the zeros of the zeta function.
- If times had been extremely hard for Turán up to 1938, then any appearance that they were about to get better was short lived for soon they became far worse.
- In 1940 Turán was sent to a labour camp, and he was in and out of various forced labour camps throughout the war.
- Another remarkable fact is that extremal graph theory, an area which Turán founded, was one of the "best ideas" that he had while in the labour camps.
- Except for the Jews in the forced-labour camps, like Turán, others were sent to the gas chambers of German concentration camps.
- Turán was liberated from the labour camp in 1944 and was able to resume teaching at the Hungarian Rabbinical Training School in Budapest.
- Before this, however, Turán was able to make international contacts which let him visit Denmark for six months, then the Institute for Advanced Study at Princeton for six months, in 1947.
- in July 1976, at the meeting on combinatorics at Orsay in Paris, V T Sós (Mrs Turán) gave me the terrible news (which she had known for six years) that Paul had leukaemia.
- She said that Paul loved life too much and with a death sentence hanging over him would not be able to live and work very well.
- As regards the latter, Turán found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.
- In 1959 Turán embarked on the preparation of a new, greatly expanded version of the book.
- Constant rewriting became necessary in the light of the new improvements and applications, and, at the time of his death in 1976, the project had still not been completed to Turán's total satisfaction.
- The book is a fitting tribute to Turán's remarkable achievements in analysis, and the editors of the manuscript deserve high praise for their efforts in bringing it to publication.
- We have mentioned some of Turán's mathematics above.
- Turán's editing was remarkable.
- But this is only a part of the editorial work Turán undertook, being on the editorial boards of Acta Arithmetica, Archiv für Mathematik, Analysis Mathematica, Compositio Mathematica, Journal of Number Theory, and essentially all Hungarian mathematical journals.
- Turán received many honours in addition to the honours which we mentioned above.
- A special issue of Acta Mathematica devoted to Paul Turán was published in 1980.

Born 18 August 1910, Budapest, Hungary. Died 26 September 1976, Budapest, Hungary.

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Group Theory, Origin Hungary

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive