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Topic: Littlewood conjecture


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In the News (Fri 26 Apr 19)

  
  Littlewood conjecture - Wikipedia, the free encyclopedia
In mathematics, the Littlewood conjecture is an open problem (as of 2004) in Diophantine approximation, posed by J.
Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed.
This in turn is a special case of a general conjecture of Margulis on Lie groups.
en.wikipedia.org /wiki/Littlewood_conjecture   (345 words)

  
 NationMaster - Encyclopedia: Twin prime conjecture
Mathematicians believe the conjecture to be true, based only on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes using Cramér's model.
Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics.
Hardy and John Littlewood) is a generalization of the twin prime conjecture.
www.nationmaster.com /encyclopedia/Twin-prime-conjecture   (1352 words)

  
 John Edensor Littlewood
John Edensor Littlewood (1885 - 1977) was a British mathematician.
Littlewood was born in Rochester in Kent, and studied at Cambridge University.
Hardy, and together they devised the Hardy - Littlewood conjecture, a strong form of the twin prime conjecture.
www.ebroadcast.com.au /lookup/encyclopedia/jo/John_Edensor_Littlewood.html   (51 words)

  
 Littlewood conjecture - Encyclopedia, History, Geography and Biography
In mathematics, the Littlewood conjecture is an open problem (as of 2006) in Diophantine approximation, posed by J.
The conjecture states something about the limes inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed.
www.arikah.com /encyclopedia/Littlewood_conjecture   (387 words)

  
 Spartanburg SC | GoUpstate.com | Spartanburg Herald-Journal
He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true then the Prime Number Theorem follows and obtained the error term.
He coined Littlewood's law, which states that individuals can expect miracles to happen to them, at the rate of about one per month.
Littlewood's inequality on bilinear forms was a forerunner of the later Grothendieck tensor norm theory.
www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=John_Edensor_Littlewood   (458 words)

  
 Admissible prime constellations
This conjecture is known as the prime k-tuple conjecture.
In the same memoir [1], Hardy and Littlewood conjectured, based on empirical evidence, that rho(x) <= pi(x) for x >= 2, where rho(x) is the largest number of primes that occur indefinitely often in an interval of length x.
In 1974, Hensley and Richards proved that the pi(x) conjecture is incompatible with the prime k-tuple conjecture (see [3] for a summary of their work, and [4, problem A9]).
www.ieeta.pt /~tos/apc.html   (1090 words)

  
 Twin prime conjecture - Definition, explanation   (Site not responding. Last check: 2007-10-19)
Mathematicians believe the conjecture to be true, based only on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes.
In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs which have a distance of 2k.
Hardy and John Littlewood), which is concerned with the distribution of twin primes, in analogy to the prime number theorem.
www.calsky.com /lexikon/en/txt/t/tw/twin_prime_conjecture_1.php   (482 words)

  
 Second Hardy-Littlewood conjecture - Wikipedia, the free encyclopedia
In number theory, the second Hardy-Littlewood conjecture concerns the number of primes in intervals.
This is probably false in general as it is inconsistent with the first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x.
If the first Hardy-Littlewood conjecture holds, then the first such k-tuple is expected for x greater than 1.5 x 10
en.wikipedia.org /wiki/Second_Hardy-Littlewood_conjecture   (189 words)

  
 Some number-theoretical constants
Matthews, A generalisation of Artin's conjecture for primitive roots, Acta Arith.
This is part of a conjectural density formula for the number of twin primes not exceeding a given bound.
This is one factor in a formula for the number of primitive points of height not exceeding a given value on a cubic surface.
www.gn-50uma.de /alula/essays/Moree/Moree.en.shtml   (1108 words)

  
 News | Gainesville.com | The Gainesville Sun | Gainesville, Fla.   (Site not responding. Last check: 2007-10-19)
He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true then the Prime Number Theorem follows and obtained the error term.
In his other work Littlewood collaborated with Raymond Paley in Fourier theory, and with Cyril Offord in combinatorial work on random sums, in developments that opened up fields still intensively studied.
Together they devised the first Hardy-Littlewood conjecture, a strong form of the twin prime conjecture, and the second Hardy-Littlewood conjecture.
www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=John_Edensor_Littlewood   (450 words)

  
 Twin prime conjecture 1 - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-19)
Look for Twin prime conjecture 1 in Wiktionary, our sister dictionary project.
Look for Twin prime conjecture 1 in the Commons, our repository for free images, music, sound, and video.
Check for Twin prime conjecture 1 in the deletion log, or visit its deletion vote page if it exists.
www.sciencedaily.com /encyclopedia/twin_prime_conjecture_1   (171 words)

  
 The Prime Glossary: Goldbach's conjecture
Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes.
Goldbach's conjecture: Every even integer n greater than two is the sum of two primes.
Among other things, they conjectured that the number of ways of writing n as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times n/(log n)
primes.utm.edu /glossary/page.php?sort=GoldbachConjecture   (681 words)

  
 Second Hardy-Littlewood conjecture - Encyclopedia, History, Geography and Biography
In number theory, the second Hardy-Littlewood conjecture concerns the number of primes in intervals.
This is probably false in general as it is inconsistent with the more likely first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x.
Second Hardy-Littlewood conjecture, References, Analytic number theory, Conjectures and Unsolved problems in mathematics.
www.arikah.net /encyclopedia/Second_Hardy-Littlewood_conjecture   (208 words)

  
 Mathematical Quotations -- L
There's a touch of the priesthood in the academic world, a sense that a scholar should not be distracted by the mundane tasks of day-to-day living.
The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.
Littlewood, J. It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group.
math.furman.edu /~mwoodard/mqs/ascquotl.html   (2374 words)

  
 math lessons - Littlewood conjecture
In mathematics, the Littlewood conjecture is a open problem (as of 2004) in Diophantine approximation, posed by J.
This is now another conjecture, expected to hold for n ≥ 3: it is stated in terms of G = SL
Einsiedler, Katok and Lindenstrauss have shown that it must have Hausdorff dimension zero; and in fact is a union of countably many compact sets of box-counting dimension zero.
www.mathdaily.com /lessons/Littlewood_conjecture   (233 words)

  
 xxos
A second twin prime conjecture states that adding a correction proportional to to a computation of Brun's constant ending with will give an estimate with error less than.
An extended form of this conjecture, sometimes called the strong twin prime conjecture or first Hardy-Littlewood conjecture, states that the number of twin primes less than or equal to x is asymptotically equal to where is the so-called twin primes constant.
This conjecture is a special case of the more general k-tuple conjecture (also known as the first Hardy-Littlewood conjecture), which corresponds to the set.
www.xxos.net /list.php?category=twins&cat_id=3000017   (130 words)

  
 Second Hardy-Littlewood conjecture   (Site not responding. Last check: 2007-10-19)
The Second Hardy-Littlewood Conjecture concerns the number of primes in intervals.
If pi(x) is the number of primes up to and including x then the conjecture states:
This is probably false in general as it is inconsistent with the first Hardy-Littlewood conjecture, but the first violation is likely to occur for very large values of x and y.
www.infomutt.com /s/se/second_hardy_littlewood_conjecture.html   (101 words)

  
 [No title]
His immediate reaction was that of course the second conjecture was the true one, and he was quite surprised when I told him that he was contradicting the prevailing view among the experts.
The infinitude of prime pairs is a "very probable conjecture", "morally certain", or what have you, but the one thing it isn't is "known".
His immediate reaction was that of course the second > conjecture was the true one, and he was quite surprised when I told him > that he was contradicting the prevailing view among the experts.
www.math.niu.edu /~rusin/known-math/96/2conjectures   (1085 words)

  
 Enumeration to 1.6*10^15 of the prime quadruplets
The validity of the Hardy-Littlewood conjecture (2.3) is critical to our estimate (4.1) of Brun's B_4 constant and the associated error bound.
However, it would appear that a more rigorous error bound must await a deeper understanding of the subtleties in the distribution of the prime quadruplets in particular, and the prime constellations in general.
Hardy and J. Littlewood, "Some problems of 'Partitio Numerorum' III: On the expression of a number as a sum of primes," Acta Math.
www.trnicely.net /quads/quads.html   (2715 words)

  
 The On-Line Encyclopedia of Integer Sequences
Hardy and Littlewood conjecture that this sequence is infinite.
G. Hardy and J. Littlewood, "Some problems of `partitio numerorum'; III: on the expression of a number as a sum of primes," Acta Mathematica, Vol.
Conjecture : a(n) is asymptotic to c*n*log(n)^2 with c around 2.9...
www.research.att.com /~njas/sequences/A080149   (197 words)

  
 The On-Line Encyclopedia of Integer Sequences
Conjecturally, a(n) is the largest number of primes that occurs on an infinite number of intervals of n consecutive integers.
The First Hardy-Littlewood conjecture (k-tuples conjecture) then implies that, for an infinitude of n, the interval [n+1, n+3159] includes 447 primes.
Conjecturally, a(n) = lim sup pi(x+n)-pi(x), where pi = A000720.
www.research.att.com /~njas/sequences/A023193   (142 words)

  
 A Generalization of the Prime Pairs / Hardy-Littlewood / Dickson / Schinzel-Sierpinski Conjectures
The prime pairs conjecture is a special case of this for the 1-tuple (2).
For example, the 3-tuple N, N+1, N+2 will assuredly have members divisible by 2 and by 3, and if N is divisible by 2, then either N or N+2 will be divisible by 4.
I find it curious that Dickson's Conjecture appeared in 1904, eighteen years previous to 1922, when (I believe) Hardy and Littlewood's article appeared.
www.d.umn.edu /~schilton/Articles/primep~1.htm   (1465 words)

  
 The least K consecutive primes such that the sum of every two of them produces a distinct number
The conjecture I will assume is a generalization of Dirichlet's primes in arithmetic progression, and is very similar to the Hardy- Littlewood conjecture.
Therefore, *if we assume the Hardy-Littlewood conjecture is true*, then there exists a set of k consecutive primes such that the sum of every two of them produces a distinct number.
D is not divisible by 67, 137, 277, or 557, so by the Hardy-Littlewood conjecture and its generalization, there are an infinite number of x values for which all four of these numbers are prime.
www.primepuzzles.net /puzzles/puzz_096.htm   (948 words)

  
 Prime Numbers :: Number Theory : Gourt   (Site not responding. Last check: 2007-10-19)
The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers.
Prime numbers have been the subject of intense research, yet some fundamental questions remain such as the Riemann hypothesis or the Goldbach conjecture, which have been open for more than a century.
The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists: When looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows well-defined laws.
www.dejavu.org /cgi-bin/get.cgi?ver=93&url=http%3A%2F%2Fscience.gourt.com%2FMath%2FNumber-Theory%2FPrime-Numbers.html   (1105 words)

  
 John Edensor Littlewood Summary
English mathematician who, in collaboration with G. Hardy, produced an acclaimed body of work, including contributions to pure mathematics and the theory of series and functions.
Littlewood became a Fellow of the Royal Society in 1915 and received the organization's Royal Medal (1929), Sylvester Medal (1943), and Copley Medal (1958).
John Edensor Littlewood at the MacTutor History of Mathematics archive.
www.bookrags.com /John_Edensor_Littlewood   (329 words)

  
 Encyclopedia   (Site not responding. Last check: 2007-10-19)
He began research under the supervision of Ernest William Barnes, working on entire functions.
In his other work Littlewood collaborated with Raymond Paley in Fourier theory, and with Offord in combinatorial work on random sums, in developments that opened up fields still intensively studied.
See also: Littlewood's conjecture, Littlewood's three principles of real analysis, Littlewood-Offord problem.
simple.seowaste.com /john_edensor_littlewood   (225 words)

  
 Puzzle 315. pn => pn-i + pi
Thomas J Engelsma has found many admissible prime tuplet patterns which will each give infinitely many counter examples if the k-tuple conjecture is true.
p(n) > 467473 it is necessary first to check the conjecture from n = 39017 to 19508 and i from 1 to 19508.
It appears conceivable (but unlikely in my opinion) that (2) is true and the second Hardy-Littlewood conjecture is false.
www.primepuzzles.net /puzzles/puzz_315.htm   (863 words)

  
 Homogeneous Flows, Moduli Spaces and Arithmetic; Program
In particular we will give a proof of the Oppenheim conjecture (we may also present Margulis' original proof which does not rely on Ratner's theorem).
The emergence of the Lagrangian flag was shown to be a consequence of the Zorich-Kontsevich conjecture, that the Lyapunov spectrum of this cocycle is simple.
The rest of the minicourse will be devoted to proving a recent result, joint with Stephane Nonnenmacher, giving a positive (and explicit) lower bound for the entropy of the limit measures.
www.claymath.org /programs/summer_school/2007/program.php   (1441 words)

  
 Citebase - On the Littlewood conjecture in fields of power series
We further discuss the Littlewood conjecture for pairs of algebraic power series.
Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
[1] B. Adamczewski and Y. Bugeaud, On the Littlewood conjecture in simultaneous Dio phantine approximation, J. London Math.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0511680   (676 words)

  
 Geometry and Topology Seminar -- Fall 2003
I will highlight a few steps of this proof and sketch in greater detail the connection to Littlewood's conjecture, and how to the deduce that the set of exceptions has Hausdorff dimension zero.
This conjecture was proved by Margulis using the methods of topological dynamics on homogeneous spaces of Lie group.
I give a gentle introduction to the ideas of the proof, derive a generalization of the Oppenheim conjecture for pairs consisting of linear and quadratic forms, and discuss open problems in the area.
www.its.caltech.edu /~manning/fall2003.html   (595 words)

  
 EMS Prizes
This goes far beyond what was known earlier about Littlewood's conjecture, and spectacularly confirms the high promise of themethods of ergodic theory in studying previously intractable problems of diophantine approximation.
Okounkov gave the first proof of the celebrated Baik-Deift-Johansson conjecture, which states that the asymptotics of random partitions distributed according to the Plancherel measure coincides with that of the eigenvalues of large Hermitian matrices.
Venjakob's work applies quite generally to towers of number fields whose Galois group is an arbitrary compact p-adic Lie group (which is not, in general, commutative), and has done much to show that a rich theory is waiting to be developed.
www.math.kth.se /4ecm/prizes.ecm.html   (1412 words)

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