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I got this information regarding Riemann matrix:

RIEMANN is a N-by-N matrix associated with the Riemann hypothesis, which is true if and only if:

 DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon>0 ('!' denotes factorial).
 where, A= Riemann matrix

with,

  A = B(2:N+1, 2:N+1), 

where

  B(i,j) = i-1 if i divides j and
           -1 otherwise.

Properties include, with M = N+1:

  1. Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M.

  2. i <= E(i) <= i+1 with at most M-SQRT(M) exceptions.

  3. All integers in the interval (M/3, M/2] are eigenvalues.

Reference: F. Roesler, Riemann's hypothesis as an eigenvalue problem, Linear Algebra and Appl., 81 (1986), pp. 153-198.

Link to the Reference Paper

Now, I have a Matlab Function which creates a 3-by-3 Riemann matrix as:

    R=riemann(3)

R=

 1    -1     1
-1     2    -1
-1    -1     3

and

 eig(R)

ans =

0.5858
2.0000
3.4142

Questions:

  1. What is "O" factor in the meaning of determinant ?

  2. How do the 3-by-3 Riemann matrix "R" created by my function concur with properties stated above ? (1st property holds good but other two ??)

P.S.: If u need to take a look at my function,please tell me.

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