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

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, -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, -1, 2, 1, -1, -1, 1, 1, 6, 1, 2, 1, -1, -15, -3, 1, -1, 2, 1, 2, -1, -1, 1, 1

Offset: 1

Author

M. F. Hasler, May 08 2018

Keywords

• sign

Comments

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.

See A304239 and A304241 - A304242 for motivation & further information.

Examples

			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).)
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.
M(39) = 0 is immediately followed by another zero, M(40) = 0, the "empty" interval between these two is ignored by definition.
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.
		

Crossrefs

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304239, A304241, A304242.

Programs

PARI

M=0; for(n=1, oo, if(m=A002321(n), abs(m)>abs(M) && M=m, M && M=print1(M", ")))

Formula

a(n) = A002321(A304239(n)).