János Pintz

The native form of this personal name is Pintz János. This article uses the Western name order.

János Pintz (December 20, 1950, Budapest)^[1] is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences.

1 Mathematical results
2 See also
3 References
4 External links

Mathematical results

With János Komlós and Endre Szemerédi disproved the Heilbronn conjecture.
With Iwaniec he proved that for sufficiently large n there is a prime between n and n+n^23/42.
Pintz gave an effective upper bound for the first number for which the Mertens conjecture fails.
He gave a O(x^2/3) upper bound for the number of those numbers that are less than x and not the sum of two primes.
With Imre Z. Ruzsa he improved a result of Linnik by showing that every sufficiently large even number is the sum of two primes and at most 8 powers of 2.
Pintz is best known for the following result that he, Daniel Goldston, and Cem Yıldırım proved in 2005:^[2]

$\liminf_{n\to\infty}\frac{p_{n%2B1}-p_n}{\log p_n}=0$

where $p_n\$ denotes the nth prime number. In other words, for every c > 0, there exist infinitely many pairs of consecutive primes p_n and p_n+1 that are closer to each other than the average distance between consecutive primes by a factor of c, i.e., p_n+1 − p_n < c log p_n.

This result was originally reported in 2003 by Dan Goldston and Cem Yıldırım but was later retracted.^[3]^[4] Then Janos Pintz joined the team and completed the proof in 2005. Later they improved this to showing that p_n+1-p_n<c√log n(log log n)² occurs infinitely often.

Further, if one assumes the Elliott-Halberstam conjecture, then one can also show that primes within 16 of each other occur infinitely often, which is nearly the twin prime conjecture.

Goldston, S. W. Graham, Pintz, and Yıldırım proved that the difference between numbers which are products of exactly 2 primes is infinitely often at most 6.^[5]

References

^ Peter Hermann, Antal Pasztor: Magyar és nemzetközi ki kicsoda, 1994
^ http://arxiv.org/abs/math/0508185
^ http://aimath.org/primegaps/
^ http://www.aimath.org/primegaps/residueerror/
^ D. Goldston, S. W. Graham, J. Pintz, C. Yıldırım: Small gaps between products of two primes, Proc. Lond. Math. Soc., 98(2007) 741–774.

External links

János Pintz's page at the Alfréd Rényi Institute of Mathematics

Persondata
Name	Pintz, Janos
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Place of death

János Pintz

Contents

Mathematical results

See also

References

External links