# What are Riemann Matrix?

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


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.

• Please don't hold/delete this post, but transfer to the mathematics or computer science site. – peterh Dec 17 '13 at 11:57
• This question appears to be off-topic because it is about mathematics – Dan Dec 17 '13 at 12:04
• Given two sequences of real numbers $(a_n)$ and $(b_n)$, we write $a_n=O(b_n)$ (read "$a_n$ is a big $O$ of $b_n$") if asymptotically $a_n \leq C\cdot b_n$, for some constant $C$ independent of $n$. More precisely : there are constants $M,C>0$ such that $a_n \leq C\cdot b_n$ for all $N\geq M$. – Thibaut Dumont Dec 17 '13 at 12:45

## 1 Answer

The O(...) in the question is not a multiplicative factor, it is referring to Big O Notation and in this case it limits how the size of the determinant increases with N.

For your second question, you have an example case of N=3 which means that M=N+1=4.

So, the second property states that

1 <= E(1)=0.5858 <= 2    (false)
2 <= E(2)=2.0000 <= 3    (true)
3 <= E(3)=3.4142 <= 4    (true)


with at most M-SQRT(M) = 4-2 = 2 exceptions. Looking at the statements above we can see that there is only a single exception, so this property is fulfilled.

The third property says that all integers in the interval (M/3, M/2] are integers. In this case the interval is (4/3, 4/2] which only contains the integer 2. This value is indeed one of the eigenvalues so this property is also fulfilled.

• oh yes, the Big O Notation.How could i forget it ???.Thanks. – user2888999 Dec 19 '13 at 13:51