Pál Turán (nonfiction)

Pál Turán.

Paul (Pál) Turán (Hungarian: [ˈpaːl ˈturaːn]; 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.

Turán worked primarily in number theory, but also did much work in analysis and graph theory.

In 1934 Turán used the Turán sieve to give a new and very simple proof of a 1917 result 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 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."

Much of Turán's work in 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–Turán equidistribution inequality.

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." Peter Frankl said about Turán. "He fell victim to Numerus clausus. Mathematicians have only paper and pen,he doesn't have anything in camp.So he created combinatorics which is not needed both thing." The field is known more briefly today as extremal graph theory. 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 Kővári–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.

Turán developed the power sum method to work on the Riemann hypothesis. The method deals with inequalities giving lower bounds for sums of a particular form.

Nonfiction cross-reference

• Forbidden graph characterization (nonfiction)
• Mathematician (nonfiction)
• Riemann hypothesis (nonfiction)

External links:

• Pál Turán @ Wikipedia