A084237 a(n) = M(10^n), where M(n) is Mertens's function.

1, -1, 1, 2, -23, -48, 212, 1037, 1928, -222, -33722, -87856, 62366, 599582, -875575, -3216373, -3195437, -21830254, -46758740, 899990187, 461113106, -3395895277, -2061910120, 62467771689

Offset: 0

Author

Robert G. Wilson v, May 15 2003

Keywords

• sign
• more

Links

• Bernardo Boncompagni, Selected values of the Mertens function.
• Eugene Kuznetsov, Computing the Mertens function on a GPU, arXiv:1108.0135 [math.NT], 2011.
• Harald A. Helfgott and Lola Thompson, Summing mu(n): a faster elementary algorithm, arXiv:2101.08773 [math.NT], 2021.
• Eric Weisstein's World of Mathematics, Mertens Function.

Crossrefs

Cf. A002321, A008683.

Programs

Mathematica

s = 0; i = 1; Do[ While[i <= 10^n, s = s + MoebiusMu[i]; i++ ]; Print[s], {n, 0, 50}]

Perl

use ntheory ":all"; say mertens(10**$) for 0..15; # _Dana Jacobsen, May 22 2015

Formula

Mertens's function: Sum_{k=1..n} mu(k), where mu = Möbius function (A008683).

a(n) = A002321(10^n).

Extensions

More terms from Eric W. Weisstein, Jun 27 2003

a(17) from Bernardo Boncompagni, Jul 06 2011

Corrected a(17) and added a(18)-a(22) from Eugene Kuznetsov, a(17)-a(19) independently confirmed by Richard Sladkey, Aug 28 2012

a(21)'s sign correction and a(23) from Helfgott and Thompson (2021) added by Amiram Eldar, May 21 2021