Function Reference: Volume 2 (F-O) - The MathWorks - #451

gallery

The Redheffer matrix has these properties:

• (n-floor(log2(n)))-1 eigenvalues equal to 1

• A real eigenvalue (the spectral radius) approximately sqrt(n)

• A negative eigenvalue approximately -sqrt(n)

• The remaining eigenvalues are provably "small."

• The Riemann hypothesis is true if and only if det(A) = U(72 ) for every epsilon >0.

Barrett and Jarvis conjecture that "the small eigenvalues all lie inside the unit circle abs(Z) = 1," and a proof of this conjecture, together with a proof that some eigenvalue tends to zero as n tends to infinity, would yield a new proof of the prime number theorem.

riemann — Matrix associated with the Riemann hypothesis

A = gallery( 'riemann ',n) returns an n-by-n matrix for which the Riemannhypothesisistrueifandonlyif

for every E > 0.

The Riemann matrix is defined by: A = B(2:n+1,2:n+1)

where B(i,j) = i-1 ifi divides j,andB(i,j) = -1 otherwise. The Riemann matrix has these properties:

• Each eigenvalue e(i) satisfies abs(e(i)) <= m-1/m,where m = n+1.

• i<=e(i) <=i+1with at most m-sqrt(m) exceptions.

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