A084234 Smallest k such that |M(k)| = n^2, where M(x) is Mertens's function A002321.

1, 31, 443, 1637, 2803, 9749, 19111, 24110, 42833, 59426, 95514, 230227, 297335, 297573, 299129, 355541, 897531, 924717, 926173, 1062397, 1761649, 1763079, 1789062, 3214693, 3218010, 3232958, 4962865, 5307549, 5343710, 6433477, 6435874, 6473791, 9990083, 10188647

Offset: 1

Author

Robert G. Wilson v, May 13 2003

Keywords

• nonn

Comments

"[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]

References

• Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, New York, 2002, page 191.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..100 (calculated using the b-file at A051402)

Crossrefs

Cf. A002321, A051402.

Programs

Mathematica

i = s = 0; Do[While[Abs[s] < n^2, s = s + MoebiusMu[i]; i++ ]; Print[i - 1], {n, 1, 25}]

Formula

a(n) = A051402(n^2). - Amiram Eldar, May 06 2024

Extensions

a(31)-a(34) from Amiram Eldar, May 06 2024