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%I #4 Mar 30 2012 17:30:54
%S 1,31,443,1637,2803,9749,19111,24110,42833,59426,95514,230227,297335,
%T 297573,299129,355541,897531,924717,926173,1062397,1761649,1763079,
%U 1789062,3214693,3218010,3232958,4962865,5307549,5343710,6433477
%N Smallest k such that |M(k)| = n^2, where M(x) is Mertens's function A002321.
%C "[I]f the absolute value of M(n) can be proved to be always less than the square root of n, then the Riemann Hypothesis is true. This is called Mertens's conjecture. ... Then along came Andrew Odlyzko and his colleague, Herman te Riele and they showed in 1984 that there is a number, far larger than 10^30, that invalidates Mertens's conjecture - call it N. In other words, M(N) is greater than the square of N. So the conjecture is not true." [Sabbagh]
%D Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, New York, 2002, page 191.
%t i = s = 0; Do[While[Abs[s] < n^2, s = s + MoebiusMu[i]; i++ ]; Print[i - 1], {n, 1, 25}]
%Y Cf. A051402.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, May 13 2003
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