login
From Mertens's conjecture (1): floor(sqrt(n)) - |M(n)|, where M is Mertens's function A002321.
5

%I #19 May 21 2023 14:49:55

%S 0,1,0,1,0,1,0,0,1,2,1,1,0,1,2,3,2,2,1,1,2,3,2,2,3,4,4,4,3,2,1,1,2,3,

%T 4,5,4,5,6,6,5,4,3,3,3,4,3,3,4,4,5,5,4,4,5,5,6,7,6,6,5,6,6,7,8,7,6,6,

%U 7,6,5,5,4,5,5,5,6,5,4,4,5,6,5,5,6,7,8,8,7,7,8,8,9

%N From Mertens's conjecture (1): floor(sqrt(n)) - |M(n)|, where M is Mertens's function A002321.

%C Mertens conjectured that |A002321(n)| < sqrt(n) for all n > 1. This is now known to be false. So eventually there will be negative terms.

%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.2.

%D K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 267.

%H Paolo Xausa, <a href="/A059571/b059571.txt">Table of n, a(n) for n = 1..10000</a>

%H A. M. Odlyzko and H. J. J. te Riele, <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">Disproof of the Mertens conjecture</a>, J. reine angew. Math., 357 (1985), pp. 138-160.

%t Table[Floor[Sqrt[n]] - Abs[Plus @@ MoebiusMu[Range[n]]], {n, 1, 80}] (* Carl Najafi, Aug 17 2011 *)

%Y Cf. A002321, A059572, A059581.

%K sign

%O 1,10

%A _N. J. A. Sloane_, Feb 16 2001