Mercurial > matrix-functions
view toolbox/redheff.m @ 2:c124219d7bfa draft
Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Thu, 07 May 2015 18:36:24 +0200 |
parents | 8f23314345f4 |
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function A = redheff(n) %REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. % A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by % A(i,j) = 1 if j = 1 or if i divides j, % A(i,j) = 0 otherwise. % It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, % a real eigenvalue (the spectral radius) approximately SQRT(N), % a negative eigenvalue approximately -SQRT(N), % and the remaining eigenvalues are provably ``small''. % Barrett and Jarvis (1992) 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. % The Riemann hypothesis is true if and only if % DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 % (`!' denotes factorial). % See also RIEMANN. % Reference: % W.W. Barrett and T.J. Jarvis, % Spectral Properties of a Matrix of Redheffer, % Linear Algebra and Appl., 162 (1992), pp. 673-683. i = (1:n)'*ones(1,n); A = ~rem(i',i); A(:,1) = ones(n,1);