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annotate toolbox/riemann.m @ 2:c124219d7bfa draft

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Re-add the 1995 toolbox after noticing the statement in the ~higham/mctoolbox/ webpage.
author Antonio Pino Robles <data.script93@gmail.com>
date Thu, 07 May 2015 18:36:24 +0200
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1 function A = riemann(n)
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2 %RIEMANN A matrix associated with the Riemann hypothesis.
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3 % A = RIEMANN(N) is an N-by-N matrix for which the
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4 % Riemann hypothesis is true if and only if
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5 % DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0
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6 % (`!' denotes factorial).
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7 % A = B(2:N+1, 2:N+1), where
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8 % B(i,j) = i-1 if i divides j and -1 otherwise.
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9 % Properties include, with M = N+1:
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10 % Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M.
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11 % i <= E(i) <= i+1 with at most M-SQRT(M) exceptions.
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12 % All integers in the interval (M/3, M/2] are eigenvalues.
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13 %
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14 % See also REDHEFF.
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15
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16 % Reference:
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17 % F. Roesler, Riemann's hypothesis as an eigenvalue problem,
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18 % Linear Algebra and Appl., 81 (1986), pp. 153-198.
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19
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20 n = n+1;
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21 i = (2:n)'*ones(1,n-1);
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22 j = i';
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23 A = i .* (~rem(j,i)) - ones(n-1);