function A = riemann(n)
%RIEMANN    A matrix associated with the Riemann hypothesis.
%           A = RIEMANN(N) is an N-by-N matrix for which the
%           Riemann hypothesis is true if and only if
%           DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0
%                                             (`!' denotes factorial).
%           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:
%              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.
%
%           See also REDHEFF.

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

n = n+1;
i = (2:n)'*ones(1,n-1);
j = i';
A = i .* (~rem(j,i)) - ones(n-1);