János Pintz (born 20 December 1950 in
Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union by population within city limits and the second-largest city on the Danube river; the city has an estimated population o ...
) is a
Hungarian mathematician working in
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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
. He is a fellow of the
Rényi Mathematical Institute and is also a member of the
Hungarian Academy of Sciences. In 2014, he received the
Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number ...
.
Mathematical results
Pintz is best known for proving in 2005 (with
Daniel Goldston and
Cem Yıldırım) that
::
where
denotes the ''n''
th prime number. In other words, for every ε > 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 ε, i.e., ''p''
''n''+1 − ''p''
''n'' < ε log ''p''
''n''. This result was originally reported in 2003 by
Daniel Goldston and
Cem Yıldırım but was later retracted.
Pintz joined the team and completed the proof in 2005. Later they improved this to showing that ''p''
''n''+1 − ''p''
''n'' < ε(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.
Additionally,
* With
János Komlós and
Endre Szemerédi
Endre Szemerédi (; born August 21, 1940) is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science a ...
he 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
In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1 ...
fails.
* He gave an 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
Yuri Vladimirovich Linnik (russian: Ю́рий Влади́мирович Ли́нник; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics.
Linnik was born in B ...
by showing that every sufficiently large even number is the sum of two primes and at most 8 powers of 2.
* 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.
[D. Goldston, S. W. Graham, J. Pintz, C. Yıldırım: Small gaps between products of two primes, Proc. Lond. Math. Soc., 98(2007) 741–774.]
See also
*
Prime gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the
''n''-th prime numbers, i.e.
:g_n = p_ - p_n.\
W ...
*
Landau's problems
*
Fazekas Mihály Gimnázium
*
Maier's theorem In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer.
The theorem states that if π is the prime-counting function and λ is greater t ...
References
External links
János Pintz's pageat the
Alfréd Rényi Institute of Mathematics
The Alfréd Rényi Institute of Mathematics ( hu, Rényi Alfréd Matematikai Kutatóintézet) is the research institute in mathematics of the Hungarian Academy of Sciences. It was created in 1950 by Alfréd Rényi, who directed it until his death. ...
*
{{DEFAULTSORT:Pintz, Janos
Number theorists
Institute for Advanced Study visiting scholars
Members of the Hungarian Academy of Sciences
Eötvös Loránd University alumni
Mathematicians from Budapest
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Living people
1950 births