www.gusucode.com > elmat工具箱matlab源码程序 > elmat/private/riemann.m
function A = riemann(n,classname) %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: % F. Roesler, Riemann's hypothesis as an eigenvalue problem, % Linear Algebra and Appl., 81 (1986), pp. 153-198. % % Nicholas J. Higham % Copyright 1984-2015 The MathWorks, Inc. n = n+1; i = (2:cast(n,classname))'; A = i .* ~rem(i', i) - 1;