function A = riemann(n)
%RIEMANN Matrix associated with the Riemann hypothesis.
%   A = GALLERY('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.
%
%   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 PRIVATE/REDHEFF.

%   Reference:
%   [1] F. Roesler, Riemann's hypothesis as an eigenvalue problem,
%       Linear Algebra and Appl., 81 (1986), pp. 153-198.
%
%   Nicholas J. Higham, Dec 1999.
%   Copyright 1984-2000 The MathWorks, Inc. 
%   $Revision: 1.8 $  $Date: 2000/06/01 16:46:19 $

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