Pál Turán
Encyclopedia
Paul Turán (ˈtuɾaːn; 18 August 1910–26 September 1976)
was a 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.
on 18 August 1910. He received a teaching degree at the University of Budapest
in 1933 and the Ph.D.
degree under Lipót Fejér
in 1935. As a victim of numerus clausus, he could not get university job for several years. He was sent to 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.
He became associate professor at the University of Budapest
in 1945 and full professor in 1949. He married mathematician Vera Sós in 1952 and they have two children.
He died in Budapest
on 26 September 1976 of leukemia.
, but also did much work in analysis
and graph theory
.
of G. H. Hardy
and Ramanujan on the 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. Halász
says "Its true significance lies in the fact that it was the starting point of probabilistic number theory
".
The Turán–Kubilius inequality
is a generalization of this work.
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 residue classes
. 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."
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-Turán equidistribution inequality.
. Turán's best-known result in this area is Turán's Graph Theorem
, that gives an upper bound on the number of edges in a graph that does not contain the complete graph
Kr 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 Turán's brick factory problem
, namely of determining the crossing number of a complete bipartite graph.
. The method deals with inequalities giving lower bounds for sums of the form
hence the name "power sum".
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.
was a Hungarian
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who worked primarily in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. He had a long collaboration with fellow Hungarian mathematician Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, lasting 46 years and resulting in 28 joint papers.
Life and education
Turán was born in BudapestBudapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
on 18 August 1910. He received a teaching degree at the University of Budapest
University of Budapest
The Eötvös Loránd University or ELTE, founded in 1635, is the largest university in Hungary, located in Budapest.-History:The university was founded in 1635 in Nagyszombat by the archbishop and theologian Péter Pázmány. Leadership was given over to the Jesuits...
in 1933 and the Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
degree under Lipót Fejér
Lipót Fejér
Lipót Fejér , was a Hungarian mathematician. Fejér was born Leopold Weiss, and changed to the Hungarian name Fejér around 1900....
in 1935. As a victim of numerus clausus, he could not get university job for several years. He was sent to labour service
Labour service (Hungary)
Labour service arose in Hungary during World War II as the required military substitution for Jewish men, who were no longer permitted to serve in the regular armed forces since the passing of the Hungarian anti-Jewish laws...
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.
He became associate professor at the University of Budapest
University of Budapest
The Eötvös Loránd University or ELTE, founded in 1635, is the largest university in Hungary, located in Budapest.-History:The university was founded in 1635 in Nagyszombat by the archbishop and theologian Péter Pázmány. Leadership was given over to the Jesuits...
in 1945 and full professor in 1949. He married mathematician Vera Sós in 1952 and they have two children.
He died in Budapest
Budapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
on 26 September 1976 of leukemia.
Work
Turán worked primarily in number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, but also did much work in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
.
Number theory
In 1934 Turán gave a new and very simple proof of a 1917 resultHardy–Ramanujan theorem
In mathematics, the Hardy–Ramanujan theorem, proved by , states that the normal order of the number ω of distinct prime factors of a number n is log...
of G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
and Ramanujan on the normal order
Normal order of an arithmetic function
In number theory, the normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values.Let ƒ be a function on the natural numbers...
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. Halász
Gábor Halász (mathematician)
Gábor Halász is a Hungarian mathematician. He mainly works is number theory and mathematical analysis, especially in analytic number theory. He is a member of the Hungarian Academy of Sciences. Since 1985, he is professor at the Eötvös Loránd University, Budapest.With Paul Turán, Halász proved...
says "Its true significance lies in the fact that it was the starting point of probabilistic number theory
Probabilistic number theory
Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions of number theory. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables...
".
The Turán–Kubilius inequality
Turán–Kubilius inequality
The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a special case in 1934 by Paul Turán and generalized in 1956 and 1964 by Jonas Kubilius.-Statement of...
is a generalization of this work.
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 residue classes
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
. With his coauthor Knapowski he proved results concerning Chebyshev's bias
Chebyshev's bias
In prime number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Chebyshev in 1853....
.
The Erdős–Turán conjecture makes a statement about primes in arithmetic progression
Primes in arithmetic progression
In number theory, the phrase primes in arithmetic progression refers to at least three prime numbers that are consecutive terms in an arithmetic progression, for example the primes ....
.
Much of Turán's number theory work dealt with the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
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."
Analysis
Much of Turán's work in analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
was tied to his number theory work. Outside of this he proved Turán's inequalities
Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by . There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors....
relating the values of the Legendre polynomials for different indices, and, together with Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
, the Erdős-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." The field is known more briefly today as extremal graph theoryExtremal graph theory
Extremal graph theory is a branch of the mathematical field of graph theory. Extremal graph theory studies extremal graphs which satisfy a certain property. Extremality can be taken with respect to different graph invariants, such as order, size or girth...
. Turán's best-known result in this area is Turán's Graph Theorem
Turán's theorem
In graph theory, Turán's theorem is a result on the number of edges in a Kr+1-free graph.An n-vertex graph that does not contain any -vertex clique may be formed by partitioning the set of vertices into r parts of equal or nearly-equal size, and connecting two vertices by an edge whenever they...
, that gives an upper bound on the number of edges in a graph that does not contain the complete graph
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
Kr as a subgraph. He invented the Turán graph
Turán graph
The Turán graph T is a graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have subsets of size \lceil n/r\rceil, and r- subsets of size \lfloor n/r\rfloor...
, a generalization of the complete bipartite graph
Complete bipartite graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.- Definition :...
, 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 Turán's brick factory problem
Crossing number (graph theory)
In graph theory, the crossing number cr of a graph G is the lowest number of edge crossings of a planar drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero.The concept originated in...
, 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 hypothesisRiemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
. The method deals with inequalities giving lower bounds for sums of the form
hence the name "power sum".
Besides its applications in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
, it has been used in function theory, numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, differential equations, transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, and estimating the number of zeroes of a function in a disk.
Honors
- Hungarian Academy of SciencesHungarian Academy of SciencesThe Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
elected corresponding member in 1948 and ordinary member in 1953 - Kossuth PrizeKossuth PrizeThe Kossuth Prize is a state-sponsored award in Hungary, named after the Hungarian politician and revolutionary Lajos Kossuth. The Prize was established in 1948 by the Hungarian National Assembly, to acknowledge outstanding personal and group achievements in the fields of...
in 1948 and 1952 - Tibor Szele Prize of János Bolyai Mathematical SocietyJános Bolyai Mathematical SocietyThe János Bolyai Mathematical Society is the Hungarian mathematical society, named after János Bolyai, a 19th century Hungarian mathematician, a co-discoverer of non-Euclidean geometry. It is the professional society of the Hungarian mathematicians, applied mathematicians, and mathematics teachers...
1975
External links
- Paul Turán memorial lectures at the Rényi Institute