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| In [[mathematics]], the '''Hilbert–Pólya conjecture''' is a possible approach to the [[Riemann hypothesis]], by means of [[spectral theory]].
| | I am Sonja from Belvidere. I am learning to play the Cello. Other hobbies are Cycling.<br><br>Feel free to visit my web blog; [http://ebook-pdfree.com/free-thirteen-reasons-pdf-download/ Thirteen Reasons Why PDF] |
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| ==History==
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| In a letter to [[Andrew Odlyzko]], dated January 3, 1982, [[George Pólya]]
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| said that while he was in [[Göttingen]] 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
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| :<math> \tfrac12 + it </math>
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| of the [[Riemann zeta function]]
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| corresponded to [[eigenvalues]] of an unbounded [[self adjoint operator]].<ref name="odlyzko">{{citation|url=http://www.dtc.umn.edu/~odlyzko/polya/index.html|last=Odlyzko|first=Andrew|authorlink=Andrew Odlyzko|title=Correspondence about the origins of the Hilbert–Polya Conjecture}}.</ref> The earliest published statement of the conjecture seems to be in {{harvtxt|Montgomery|1973}}.<ref name="odlyzko"/><ref name="montgomery">{{Citation | last1=Montgomery | first1=Hugh L. | title=Analytic number theory |series=Proc. Sympos. Pure Math.|volume= XXIV | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr = 0337821 | year=1973 | chapter=The pair correlation of zeros of the zeta function | pages=181–193}}.</ref>
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| ===1950s and the Selberg trace formula===
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| At the time of Pólya's conversation with Landau, there was little basis for such speculation. However [[Atle Selberg|Selberg]] in the early 1950s proved a duality between the length [[spectrum]] of a [[Riemann surface]] and the [[eigenvalue]]s of its [[Laplacian]]. This so-called [[Selberg trace formula]] bore a striking resemblance to the [[explicit formulae (L-function)|explicit formulae]], which gave credibility to the speculation of Hilbert and Pólya.
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| ===1970s and random matrices===
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| [[Hugh Montgomery (mathematician)|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.<ref name="montgomery"/> 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]].
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| Dyson saw that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random [[Hermitian matrix]]. These distributions are of importance in physics — the [[eigenstate]]s of a [[Hamiltonian (quantum mechanics)|Hamiltonian]], for example the [[energy level]]s of an [[atomic nucleus]], satisfy such statistics. Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the [[Gaussian unitary ensemble]], and both are now believed to obey the same statistics. Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.<ref>{{citation|first1=Zeev|last1=Rudnick|first2=Peter|last2=Sarnak|author2-link=Peter Sarnak|year=1996|url=http://www.math.tau.ac.il/~rudnick/papers/zeta.dvi.gz|title=Zeros of Principal L-functions and Random Matrix Theory|journal=Duke Journal of Mathematics|volume=81|pages=269–322}}.</ref>
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| ===Recent times===
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| In a development that has given substantive force to this approach to the Riemann hypothesis through [[functional analysis]], [[Alain Connes]] has formulated a [[trace formula]] that is actually equivalent to the [[Riemann hypothesis]]. This has therefore strengthened the analogy with the [[Selberg trace formula]] to the point where it gives precise statements. He gives a geometric interpretation of the [[Explicit formulae (L-function)|explicit formula]] of number theory as a trace formula on [[noncommutative geometry]] of [[adele ring|Adele]] classes.<ref>{{citation|arxiv=math/9811068|last=Connes|first=Alain|authorlink=Alain Connes|title=Trace formula in noncommutative geometry and the zeros of the Riemann zeta function|year=1998}}.</ref>
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| ==Possible connection with quantum mechanics==
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| A possible connection of Hilbert–Pólya operator with [[quantum mechanics]] was given by Pólya. The Hilbert–Pólya conjecture operator is of the form <math>\scriptstyle 1/2+iH</math> where <math>\scriptstyle H</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of a particle of mass <math>m</math> that is moving under the influence of a potential <math>\scriptstyle V(x)</math>. The Riemann conjecture is equivalent to the assertion that the Hamiltonian is [[Hermitian operator|Hermitian]], or equivalently that <math>\scriptstyle V</math> is real.
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| Using [[Perturbation theory (quantum mechanics)|perturbation theory]] to first order, the energy of the ''n''th eigenstate is related to the [[expectation value]] of the potential:
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| :<math> E_{n}=E_{n}^{0}+ \langle \varphi^{0}_n \vert V \vert \varphi^{0}_n \rangle </math>
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| where <math>\scriptstyle E^{0}_n</math> and <math>\scriptstyle \varphi^{0}_n</math> 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 <math>\scriptstyle E_n</math>. Such integral equations may be solved by means of the [[resolvent kernel]], so that the potential may be written as
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| :<math> V(x)=A\int_{-\infty}^{\infty} (g(k)+\overline{g(k)}-E_{k}^{0})\,R(x,k)\,dk </math>
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| where <math>\scriptstyle R(x,k)</math> is the resolvent kernel, <math>\scriptstyle A</math> is a real constant and
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| :<math> g(k)=i \sum_{n=0}^{\infty} \left(\frac{1}{2}-\rho_n \right)\delta(k-n) </math>
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| where <math>\scriptstyle \delta(k-n)</math> is the [[Dirac delta function]], and the <math>\scriptstyle \rho_n</math> are the "non-trivial" roots of the zeta function <math>\scriptstyle \zeta (\rho_n)=0 </math>.
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| [[Michael Berry (physicist)|Michael Berry]] and Jon Keating have speculated that the Hamiltonian ''H'' is actually some [[Quantization (physics)|quantization]] of the classical Hamiltonian ''xp'', where ''p'' is the [[canonical momentum]] associated with ''x''<ref name="bk99a">{{Citation | last1=Berry | first1=Michael V. | author1-link = Michael Berry (physicist) | last2=Keating | first2=Jonathan P. | editor1-last=Keating | editor1-first=Jonathan P. | editor2-last=Khmelnitski | editor2-first=David E. | editor3-last=Lerner | editor3-first=Igor V. | title=Supersymmetry and Trace Formulae: Chaos and Disorder | url=http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry306.pdf | publisher=Plenum | location=New York | isbn=978-0-306-45933-7 | year=1999a | chapter=H = xp and the Riemann zeros | pages=355–367}}.</ref> The simplest Hermitian operator corresponding to ''xp'' is
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| :<math>H = \tfrac1{2} (xp+px) = - i \left( x \frac{\mathrm{d}}{\mathrm{d} x} + \frac1{2} \right).</math>
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| This refinement of the Hilbert–Pólya conjecture is known as the ''Berry conjecture'' (or the ''Berry–Keating conjecture''). As of 2008, it is still quite inconcrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Keating have conjectured that since this operator is invariant under [[Dilation (operator theory)|dilation]]s perhaps the boundary condition ''f''(''nx'') = ''f''(''x'') for integer 'n' may help to get the correct asymptotic results valid for big 'n'
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| :<math> \frac{1}{2} + i \frac{ 2\pi n}{\log n}. </math><ref>{{citation| last1=Berry | first1=Michael V. | author1-link = Michael Berry (physicist) | last2=Keating | first2=Jonathan P. | year=1999b | url=http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry307.pdf|title=The Riemann zeros and eigenvalue asymptotics|journal=SIAM Review|volume= 41|issue=2|pages=236–266}}.</ref>
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| ==References==
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| {{reflist}}
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| ==Additional reading==
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| * {{citation|last=Aneva|first=B.|url=http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/aneva.pdf|title=Symmetry of the Riemann operator|year=1999|journal=Physics Letters|volume= B450|pages= 388–396}}.
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| *{{citation|last=Elizalde|first=Emilio|title=Zeta regularization techniques with applications|isbn=978-981-02-1441-8|publisher=World Scientific|year=1994}}. Here the author explains in what sense the problem of Hilbert–Polya is related with the problem of the Gutzwiller trace formula and what would be the value of the sum <math> \exp(i\gamma) </math> taken over the imaginary parts of the zeros.
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| {{DEFAULTSORT:Hilbert-Polya conjecture}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Conjectures]]
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I am Sonja from Belvidere. I am learning to play the Cello. Other hobbies are Cycling.
Feel free to visit my web blog; Thirteen Reasons Why PDF