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{{Infobox scientist
|name              =  Paul (Pál) Turán
|image = Bundesarchiv Bild 183-33149-0001, Leipzig, Universität, Professor Turan.jpg
|image_size = 150px
|caption          =
|birth_date        = {{birth date|1910|8|18|df=y}}
|birth_place      = [[Budapest]], [[Kingdom of Hungary|Hungary]]
|death_date        = {{death date and age| 1976 |9|26|1910|8|18|df=y}}
|death_place      = [[Budapest]], [[People's Republic of Hungary|Hungary]]
|residence        = [[Hungary]]
|citizenship      =
|nationality      = [[Hungary|Hungarian]]
|ethnicity        =
|fields            = [[Mathematics]]
|workplaces        = [[University of Budapest]]
|alma_mater        = [[University of Budapest]]
|doctoral_advisor  = [[Lipót Fejér]]
|academic_advisors =
|doctoral_students = [[László Babai]]
|notable_students  =
|known_for        = Power sum method<br>[[Extremal graph theory]]
|author_abbrev_bot =
|author_abbrev_zoo =
|influences        =
|influenced        =
|awards            = [[Kossuth Prize]]<br>Tibor Szele Prize
|religion          =
|signature        =  <!--(filename only)-->
|footnotes        =
}}
{{eastern name order|Turán Pál}}
'''Paul (Pál) Turán''' ({{IPA-hu|ˈturaːn|lang}}; 18 August 1910 &ndash; 26 September 1976)<ref name="JNT obit">
{{cite journal
| last = Alpár
| first = L.
|date=August 1981
| title = In memory of Paul Turán
| journal = Journal of Number Theory
| volume = 13
| issue = 3
| pages = 271–278
| publisher = Academic Press
| doi = 10.1016/0022-314X(81)90012-3
}}
</ref>{{Rp|271}}<ref name="Hungarian biography">
{{cite web
|url= http://mek.oszk.hu/00300/00355/html/ABC15363/16089.htm
|title= Magyar Életrajzi Lexikon: Turán Pál
|accessdate= 2008-06-21
|publisher = Magyar Elecktronikus Könyvtár (Hungarian Electronic Library)
|language= Hungarian
}}
</ref>
was a [[Hungary|Hungarian]] [[mathematician]] who worked primarily in [[number theory]]. He had a long collaboration with fellow Hungarian mathematician [[Paul Erdős]], lasting 46 years and resulting in 28 joint papers.<ref name="Erdos JAT">
{{cite journal
| last = Erdős
| first = Paul
| authorlink = Paul Erdős
| year = 1980
| title = Some notes on Turán's mathematical work
| journal = Journal of Approximation Theory
| volume = 29
| issue = 1
| pages = 2–6
| url = http://www.renyi.hu/~p_erdos/1980-42.pdf
| accessdate = 2008-06-22
| doi = 10.1016/0021-9045(80)90133-1
}}


</ref>


== Life and education ==
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Turán was born in [[Budapest]] on 18 August 1910.<ref name="JNT obit" />{{Rp|271}} He received a teaching degree at the [[University of Budapest]] in 1933 and the [[Ph.D.]] degree under [[Lipót Fejér]] in 1935.<ref name="JNT obit" />{{Rp|271}} As a victim of [[Numerus clausus#Numerus clausus in Hungary|numerus clausus]], he could not get university job for several years. He was sent to [[Labour service (Hungary)|labour service]] at various times from 1940 to 1944. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work.<ref>"An officer was standing nearby, watching us work. When he heard my name, he asked the comrade whether I was a mathematician. It turned out, that the officer, Joshef Winkler, was an engineer.  In his youth, he had placed in a mathematical competition; in civilian life he was a proof-reader at the print shop where the periodical of the Third Class of the Academy (Mathematical and Natural sciences) was printed. There he had seen some of my manuscripts." P. Turán, "A note of welcome", [[Journal of Graph Theory]] '''1''' (1977), pp. 7-9.</ref>
 
He became associate professor at the [[University of Budapest]] in 1945 and full professor in 1949.<ref name="JNT obit" />{{Rp|272}} He married mathematician [[Vera Sós]] in 1952 and they had two children.<ref>
{{cite web
|url= http://www.cs.uchicago.edu/files/tr_authentic/TR-2001-03.ps
|title= In and Out of Hungary: Paul  Erdős, His Friends, and Times
|accessdate= 2008-06-22
|last= Babai
|first= László
|authorlink= László Babai
|year= 2001
|format= PostScript
|publisher= University of Chicago
}}
</ref>{{Rp|20}}
 
He died in [[Budapest]] on 26 September 1976<ref name="JNT obit" />{{Rp|271}} of leukemia.<ref name="Erdos AA obit">
{{cite journal
| last = Erdős
| first = Paul
| authorlink = Paul Erdős
| year = 1980
| title = Some personal reminiscences of the mathematical work of Paul Turán
| journal = Acta Arithmetica
| volume = 37
| issue =
| pages = 3–8
| issn = 0065-1036
| url = http://www.renyi.hu/~p_erdos/1980-43.pdf
| accessdate = 2008-06-22
}}
</ref>{{Rp|8}}
 
== Work ==
 
Turán worked primarily in [[number theory]],<ref name="Erdos AA obit" />{{Rp|4}} but also did much work in [[Mathematical analysis|analysis]] and [[graph theory]].
 
===Number theory===
In 1934 Turán gave a new and very simple proof of a 1917 [[Hardy–Ramanujan theorem|result]] of [[G. H. Hardy]] and [[Ramanujan]] on the [[normal order of an arithmetic function|normal order]] of the number of distinct prime divisors of a number ''n'', namely that it is very close to ln ln ''n''. In probabilistic terms he estimated the variance from ln ln ''n''. [[Gábor Halász (mathematician)|Halász]] says "Its true significance lies in the fact that it was the starting point of [[probabilistic number theory]]".
<ref name="Halasz AA obit">
{{cite journal
| last = Halász
| first = G.
| year = 1980
| title = The number-theoretic work of Paul Turán
| journal = Acta Arithmetica
| volume = 37
| pages = 9–19
| issn = 0065-1036
| url = http://www.numbertheory.org/obituaries/AA/turan/turan_halasz/index.html
| accessdate = 2008-06-22
}} {{Dead link|date=September 2010|bot=H3llBot}}
</ref>{{Rp|16}}
The [[Turán–Kubilius inequality]] is a generalization of this work.<ref name="Erdos AA obit" />{{Rp|5}} <ref name="Halasz AA obit" />{{Rp|16}}
 
Turán was very interested in the distribution of primes in arithmetic progressions, and he coined the term "prime number race" for irregularities in the [[distribution of prime numbers]] among [[Modular arithmetic|residue classes]].<ref name="Erdos AA obit" />{{Rp|5}} With his coauthor Knapowski he proved results concerning [[Chebyshev's bias]].
 
The Erdős–Turán conjecture makes a statement about [[primes in arithmetic progression]].
 
Much of Turán's number theory work dealt with the [[Riemann hypothesis]] and he developed the power sum method (see below) to help with this. Erdős said "Turán was an 'unbeliever,' in fact, a 'pagan': he did not believe in the truth of Riemann's hypothesis."<ref name="Erdos JAT" />{{Rp|3}}
 
===Analysis===
Much of Turán's work in [[Mathematical analysis|analysis]] was tied to his number theory work. Outside of this he proved [[Turán's inequalities]] relating the values of the [[Legendre polynomials]] for different indices, and, together with [[Paul Erdős]], the [[Erdős&ndash;Turán inequality|Erdős&ndash;Turán equidistribution inequality]].
 
===Graph theory===
 
Erdős wrote of Turán, "In 1940–1941 he created the area of extremal problems in graph theory which is now one of the fastest-growing subjects in combinatorics."<ref name="Erdos JAT" />{{Rp|4}} The field is known more briefly today as [[extremal graph theory]]. Turán's best-known result in this area is [[Turán's theorem|Turán's Graph Theorem]], that gives an upper bound on the number of edges in a graph that does not contain the [[complete graph]] ''K<sub>r</sub>'' as a subgraph. He invented the [[Turán graph]], a generalization of the [[complete bipartite graph]], to prove his theorem. He is also known for the [[Kövari–Sós–Turán theorem]] bounding the number of edges that can exist in a bipartite graph with certain forbidden subgraphs,
and for raising [[Crossing number (graph theory)|Turán's brick factory problem]], namely of determining the crossing number of a complete bipartite graph.
 
===Power sum method===
 
Turán developed the power sum method to work on the [[Riemann hypothesis]].<ref name="Halasz AA obit" />{{Rp|9–14}} The method deals with inequalities giving lower bounds for sums of the form
:<math> \max_{\nu=m+1,\dots,m+n} \left | \sum_{j=1}^n b_j z_j^\nu \right |, </math>
hence the name "power sum".<ref name="Tijdeman review">
{{cite journal
| last = Tijdeman
| first = R.
| authorlink = Robert Tijdeman
|date=April 1986
| title = Book reviews: On a new method of analysis and its applications
| journal = Bulletin of the American Mathematical Society
| volume = 14
| issue = 2
| pages = 318–322
| publisher = American Mathematical Society
| location = Providence, RI
  | url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183553181
| format = PDF
| accessdate = 2008-06-22
| doi = 10.1090/S0273-0979-1986-15456-X
}}
</ref>{{Rp|319}}
Besides its applications in [[analytic number theory]], it has been used in [[function theory]], [[numerical analysis]], [[differential equations]], [[transcendence theory]], and estimating the number of zeroes of a function in a disk.<ref name="Tijdeman review" />{{Rp|320}}
 
== Publications ==
 
* {{cite book | title = Number Theory | publisher = North-Holland Pub. Co | location = Amsterdam | year = 1970 | isbn = 978-0-7204-2037-1 | author = Ed. by P. Turán. }}
* {{cite book | title = On a New Method of Analysis and Its Applications | publisher = Wiley-Interscience | location = New York | year = 1984 | isbn = 978-0-471-89255-7 | author = Paul Turán }} Deals with the power sum method.
* {{cite book | title = Collected Papers of Paul Turán | publisher = Akadémiai Kiadó | location = Budapest | year = 1990 | isbn = 978-963-05-4298-2 | author = edited by Paul Erdős. }}
 
== Honors ==
 
* [[Hungarian Academy of Sciences]] elected corresponding member in 1948 and ordinary member in 1953<ref name="JNT obit" />{{Rp|272}}
* [[Kossuth Prize]] in 1948 and 1952<ref name="JNT obit" />{{Rp|272}}
* Tibor Szele Prize of [[János Bolyai Mathematical Society]] 1975<ref name="JNT obit" />{{Rp|272}}
 
==Notes==
{{Reflist}}
 
==External links==
*{{MathGenealogy |id=37275}}
*{{MacTutor Biography|id=Turan|title=Paul Turán}}
* [http://www.renyi.hu/turanlectures_vk.html Paul Turán memorial lectures] at the Rényi Institute
 
{{Authority control|VIAF=24641014}}
{{Persondata
|NAME              = Turán, Pál
|ALTERNATIVE NAMES = Turán, Paul
|SHORT DESCRIPTION = Mathematician
|DATE OF BIRTH    = 28 August 1910
|PLACE OF BIRTH    = [[Budapest]], [[Hungary]]
|DATE OF DEATH    = 26 September 1976
|PLACE OF DEATH    = [[Budapest]], [[Hungary]]
}}
{{DEFAULTSORT:Turan, Pal}}
[[Category:1910 births]]
[[Category:1976 deaths]]
[[Category:20th-century mathematicians]]
[[Category:Number theorists]]
[[Category:Combinatorialists]]
[[Category:Hungarian mathematicians]]
[[Category:Members of the Hungarian Academy of Sciences]]
[[Category:Hungarian Jews]]
[[Category:Graph theorists]]

Latest revision as of 10:44, 10 December 2014


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