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
A = B(2:N+1, 2:N+1),
B(i,j) = i-1 if i divides j and -1 otherwise.
Properties include, with M = N+1:
- Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M.
- i <= E(i) <= i+1 with at most M-SQRT(M) exceptions.
- 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.
Now, I have a Matlab Function which creates a 3-by-3 Riemann matrix as:
1 -1 1 -1 2 -1 -1 -1 3
0.5858 2.0000 3.4142
- What is "O" factor in the meaning of determinant ?
- 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.