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Primes as Energy levels (eigenvalues of a certain operator)

  1. Jul 16, 2006 #1
    "primes" as Energy levels...(eigenvalues of a certain operator)

    I have heard about the Riemann Zeta function to be some kind of physical partition function..my question is..could we consider primes as "Energy levels" (eigenvalues) of a certain partition function or operator?..in the form that exit an operator P so:

    [tex] P|\psi>=p_{n}|\psi> [/tex]

    could someone give some information of the "Riemann zeta function" as an statistical partition function?..thanks.
     
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  3. Jul 16, 2006 #2

    matt grime

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    It's called random matrix theory, it is very famous, I've explained it to you before, and you can use google.
     
  4. Jul 16, 2006 #3
    My question is...could be the sum:

    [tex] \sum_{n} e^{-sp_{n}}=f(s) [/tex] be interpreted as the "Trace" of certain operator so we can give an "estimation" for this SUm f(s), consierng primes are "eigenvalues" of a certain Hermitian operator that have a random matrix approach, or if it is satisfied that if we have:

    [tex] P|\psi>=p_n |\psi> [/tex]

    then [tex] f(s)=Tr[e^{-sP}] [/tex] (at least as an approximation)
     
  5. Jul 17, 2006 #4

    CRGreathouse

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    What if eljose gave a thread and no one came?
     
  6. Nov 7, 2011 #5
    Re: "primes" as Energy levels...(eigenvalues of a certain operator)

    I don't know of any matrix that has all the primes as its only eigenvalues. But there appears to be a matrix such that its most negative eigenvalue (one eigenvalue per matrix) is a prime plus minus a small number.

    [itex] T(n,1)=1, T(1,k)=1, n>=k: -\sum\limits_{i=1}^{k-1} T(n-i,k), n<k: -\sum\limits_{i=1}^{n-1} T(k-i,n)[/itex]


    [itex]\displaystyle T(n,k) = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \end{bmatrix}[/itex]

    For which the list of the most negative eigenvalue for the first 100 matrices is:

    {-1., 1.41421, 2.65544, 3.43931, 4.77106, 5.24392, 6.84437, 7.15539, \
    7.47476, 7.57341, 10.9223, 11.096, 12.9021, 13.0453, 13.259, 13.4055, \
    16.9724, 17.0824, 18.9443, 19.0552, 19.2282, 19.307, 22.9972, \
    23.0759, 23.1576, 23.2173, 23.2976, 23.3972, 29.0103, 29.0407, \
    30.963, 31.0104, 31.1008, 31.1505, 31.268, 31.34, 37.0284, 37.0658, \
    37.1289, 37.174, 41.029, 41.0503, 42.9921, 43.0326, 43.0807, 43.1149, \
    46.996, 47.0293, 47.0619, 47.1025, 47.1582, 47.2011, 53.0192, \
    53.0497, 53.1076, 53.1419, 53.1893, 53.2117, 59.0477, 59.0681, \
    61.0248, 61.0474, 61.0812, 61.1071, 61.1644, 61.1812, 67.0341, \
    67.059, 67.0929, 67.1062, 71.027, 71.0496, 73.014, 73.0331, 73.0575, \
    73.0829, 73.1282, 73.1427, 79.024, 79.0442, 79.0633, 79.0799, \
    83.0154, 83.0287, 83.0648, 83.0806, 83.1091, 83.1312, 89.032, \
    89.0463, 89.0784, 89.0973, 89.1237, 89.1374, 89.1731, 89.1921, \
    97.0597, 97.0753, 97.0963, 97.1128}

    which when rounded is:

    {-1, 1, 3, 3, 5, 5, 7, 7, 7, 8, 11, 11, 13, 13, 13, 13, 17, 17, 19, \
    19, 19, 19, 23, 23, 23, 23, 23, 23, 29, 29, 31, 31, 31, 31, 31, 31, \
    37, 37, 37, 37, 41, 41, 43, 43, 43, 43, 47, 47, 47, 47, 47, 47, 53, \
    53, 53, 53, 53, 53, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, \
    71, 71, 73, 73, 73, 73, 73, 73, 79, 79, 79, 79, 83, 83, 83, 83, 83, \
    83, 89, 89, 89, 89, 89, 89, 89, 89, 97, 97, 97, 97}

    Compare this to the previous prime with the Mathematica command:

    http://www.wolframalpha.com/input/?i=Table%5BNextPrime%5Bi%2C+-1%5D%2C+%7Bi%2C+1%2C+101%7D%5D

    https://oeis.org/A191898
     

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    Last edited: Nov 7, 2011
  7. Nov 12, 2011 #6
    Re: "primes" as Energy levels...(eigenvalues of a certain operator)

    Again consider the same infinite matrix above starting:

    [itex] T(n,k) = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \end{bmatrix}[/itex]

    It then appears that the eigenvalues of a infinitely large matrix [itex] T(n,k) [/itex], when sorted and rounded, contains a infinitely long sequence of consecutive prime numbers.

    Example: The 11 most negative eigenvalues of a 300 times 300 [itex] T(n,k) [/itex] matrix are approximately: -293.072, -283.13, -281.127, -277.148, -271.195, -269.177, -263.223, -257.262, -251.299, -241.477, -239.354 which when rounded are: -293, -283, -281, -277, -271, -269, -263, -257, -251, -241, -239 which are the 52nd to the 62nd primes.
     
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