Polya Conjecture

[Gwenda Arguin]

Ina letter to Andrew Odlyzko, dated January 3, 1982, George Plyasaid that while he was in Gttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts t of the zeros

Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called Montgomery's pair correlation conjecture. The zeros tend not to cluster too closely together, but to repel.[2] Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices.

In 1998, Alain Connes formulated a trace formula that is actually equivalent to the Riemann hypothesis. This strengthened the analogy with the Selberg trace formula to the point where it gives precise statements. He gives a geometric interpretation of the explicit formula of number theory as a trace formula on noncommutative geometry of Adele classes.[8]

where E n 0 \displaystyle E_n^0 and φ n 0 \displaystyle \varphi _n^0 are the eigenvalues and eigenstates of the free particle Hamiltonian. This equation can be taken to be a Fredholm integral equation of first kind, with the energies E n \displaystyle E_n . Such integral equations may be solved by means of the resolvent kernel, so that the potential may be written as

Michael Berry and Jonathan Keating have speculated that the Hamiltonian H is actually some quantization of the classical Hamiltonian xp, where p is the canonical momentum associated with x[9] The simplest Hermitian operator corresponding to xp is

In number theory, the Plya conjecture (or Plya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Plya in 1919,[1] and proved false in 1958 by C. Brian Haselgrove. Though mathematicians typically refer to this statement as the Plya conjecture, Plya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the Riemann hypothesis. For this reason, it is more accurately called "Plya's problem".

The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general,[2] providing an illustration of the strong law of small numbers.

As far as I know, the best known result is due to Li and Yau. They proved the conjecture in the sense of "average": $\displaystyle \sum_j=1^k \lambda_j \geq \fracnW_nn+2k^(n+2)/nV(\Omega)^-n/2$.

I find their argument is elementary, only employing some standard Fourier tricks. And the big picture is quite clear if put into the quantum framework. My question is in some sense "soft", but it does make me feel absurd: what makes it so difficult to estimate the eigenvalues one by one while their average is so well understood? Does anyone work on this problem by carrying further Li & Yau's analysis? I do know one instance: Krger has transplanted their proof to Neumann settings, but how about the original Dirichlet problem?

Now why can we not do better? One of the main problems, is that you are only allowed to plugin holomorphic $f$, so it will be hard to estimate single objects. On the other hand, if you would be allowed to pluggin something compactly supported, then $\widehatf$ becomes entire, and is not compactly supported.

The fact that the support of $f$ and $\widehatf$ can not be simultaneously small, is a first instance of the uncertainity principle of Fourier analysis. The name is certainly derived from the Heisenberg uncertainity prinicple, eigenvalues are here "waves" and length of closed geodesics here "particles".

Similar things are happening for prime numbers and zeros of the Riemann zeta function (Weil's excplicit formula) or in quantum chaos (Gutzwiller trace formula). Already in finite group theory, if you try to compare traces of irreducible representations with conjugacy classes.

Let be a positive integer and the number of (not necessarily distinct) prime factors of (with ). Let be the number of positive integers with an odd number of prime factors, and the number of positive integers with an even number of prime factors. Plya (1919) conjectured that

Is it possible to deduce the shape of a drum from the sounds it makes? This is the kind of question that Iosif Polterovich, a professor in the Department of Mathematics and Statistics at Universit de Montral, likes to ask. Polterovich uses spectral geometry, a branch of mathematics, to understand physical phenomena involving wave propagation.

Plya himself confirmed his conjecture in 1961 for domains that tile a plane, such as triangles and rectangles. Until last year, the conjecture was known only for these cases. The disk, despite its apparent simplicity, remained elusive.

"Imagine an infinite floor covered with tiles of the same shape that fit together to fill the space," Polterovich said. "It can be tiled with squares or triangles, but not with disks. A disk is actually not a good shape for tiling."

Polya's conjecture is a mathematical conjecture that suppose that the sum of the first (-1)^(Omega(n)) where Omega(n) is the number of prime divisors of n with multiplicity, is always negative or zero.

A counter example is 906316571, was found fifty years ago. I wonder how could they found it because it takes a massive amount of time, I tried to optimize my python algorithm but it still takes a massive time, Can you help me optimize it ?

As chepner told you, the original year 1958 proof was not done with brute force. Neither did it reveal the smallest number to break the rule, it was only found in 1980. I have not studied the case at all, but the 1980 proof may be have been done with a computer. It is more a question of the amount of RAM available, not a question of processing speed as such.

However, with modern computers it should not be overtly difficult to approach the problem with brute force. Python is not the best alternative here, but still the numbers can be found in a reasonable time.

This is not the fastest or the most optimized function, the mathematics behind this should be quite obvious. In my machine (i7) running on one core it takes approximately 2800 seconds to calculate the full table of number of prime factors up to 1 x 10^9. (But beware, do not try this without 64-bit python and at least 8 GB RAM. The cumulative sum table consumes 4 GB.)

Due to some problems with the first numbers, the results given by the code above are slightly off. To get the official summatory Liouville lambda, use cumulative[n-1] + 2. For the number mentioned in the question (906 316 571) the result is cumulative[906316570] + 2 equalling to 829, which is the maximum value in the region.

I have finally memorized one simple mathematical statement that is true for pretty much any number you can think of, but is in fact verifyably false, with known counterexamples. The Plya conjecture is true for any number up to about 900 million, but is false for the next 300,000 or so numbers, and is ultimately false in general.

This can be explained visually as follows: Cedric and Royce are standing side by side. At time \(n\), Cedric takes a step forward if \(n\) has an even number of prime factors, and Royce steps forward if \(n\) has an odd number of prime factors. The Plya conjecture is equivalent to the statement that Royce will always be ahead of Cedric after the starting time.

En thorie des nombres, la conjecture de Plya nonce que la plupart (c'est--dire plus de la moiti) des entiers naturels infrieurs un entier donn ont un nombre impair de facteurs premiers. La conjecture a t propose par le mathmaticien hongrois George Plya en 1919[1]. En 1958, il a t prouv que celle-ci tait fausse. La taille du plus petit contre-exemple est souvent utilise pour montrer qu'une conjecture peut tre vraie pour beaucoup de nombres tout en tant fausse.

(opens in new tab)It was almost a year ago, in this space (opens in new tab), that you might have learned the astounding news that a team of two researchers from Yale University and one from Microsoft Research had announced a proof of a riddle that had eluded mathematicians for more than half a century.

The Kadison-Singer conjecture, first proposed by Richard Kadison and Isadore Singer in 1959, pertains to the mathematical foundations of quantum mechanics. At the time, experts suggested that the implications could be significant. That, says Nikhil Srivastava (opens in new tab) of Microsoft Research India (opens in new tab), is starting to come true.

Now, during the 2014 annual meeting of the Society for Industrial and Applied Mathematics (opens in new tab) (SIAM), being held in Chicago from July 7 to 11, the breakthrough is earning a more immediate reward. The 2014 George Plya Prize (opens in new tab) will be presented to Srivastava and colleagues Adam W. Marcus (opens in new tab) and Daniel A. Spielman (opens in new tab) by Irene Fonseca, professor of mathematics at Carnegie Mellon University and current SIAM president.

Plya (1887-1985) was a Hungarian mathematician who served as a professor for four decades, first at ETH Zurich, then at Stanford University. He is credited with fundamental advances in combinatorics, numerical analysis, number theory, and probability theory.

In a post written by Srivastava (opens in new tab) on the Windows on Theory blog shortly after the conjecture was proved, he emphasized the discrepancy-theoretic nature of the new result and explained its application for partitioning graphs into expanders.

For the number field case, the most developed conjecture that I know of about what might be the right sort of cohomology theory is due to Christopher Deninger. He has a very interesting recent review article about this, see also his lecture at the 1998 ICM.

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