cp's OEIS Frontend

A304240 Extremum of the Mertens function A002321 between two successive (but not adjacent) zeros.

%I A304240 #4 May 09 2018 22:57:32
%S A304240 1,-4,-3,-2,-4,2,-6,1,-2,1,-1,-8,5,-1,1,-3,-8,1,3,-1,-1,-1,1,-3,2,-1,
%T A304240 -1,-2,2,-1,-1,-9,1,7,-5,-12,-1,-2,1,-1,3,1,3,-4,1,-3,2,2,-1,-1,-1,-1,
%U A304240 -1,2,1,-1,-1,1,1,6,1,2,1,-1,-15,-3,1,-1,2,1,2,-1,-1,1,1
%N A304240 Extremum of the Mertens function A002321 between two successive (but not adjacent) zeros.
%C A304240 In view of its definition, the Mertens function A002321 does not change sign between two successive zeros. Here we list the extrema, i.e., smallest or largest value, depending on the respective sign, between two zeros, excluding the case where these zeros are immediately adjacent, i.e., A002321(k) = A002321(k+1) = 0.
%C A304240 See A304239 and A304241 - A304242 for motivation & further information.
%F A304240 a(n) = A002321(A304239(n)).
%e A304240 The Mertens function M = A002321 is defined as partial sums of the Möbius function mu. At n = 1 it has the nonzero value M(1) = 1, and at n = 2 it has its first zero, M(2) = 0. Therefore we let a(1) = 1 by convention. (One can also consider that M(0) = 0, the empty sum, is an "initial zero" preceding M(1).)
%e A304240 Between the first and second zero of M = A002321, M(2) = 0 and M(39) = 0, M takes only negative values, and the largest in absolute value is a(2) = -4.
%e A304240 M(39) = 0 is immediately followed by another zero, M(40) = 0, the "empty" interval between these two is ignored by definition.
%e A304240 The next zero is at n = 58. Between n = 40 and n = 58 M takes only negative values, and the minimum is a(3) = -3.
%o A304240 (PARI) M=0; for(n=1, oo, if(m=A002321(n), abs(m)>abs(M) && M=m, M && M=print1(M", ")))
%Y A304240 Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).
%Y A304240 Cf. A304239, A304241, A304242.
%K A304240 sign
%O A304240 1,2
%A A304240 _M. F. Hasler_, May 08 2018