A304242 Increasingly larger (in absolute value) extrema of the Mertens function A002321 between subsequent zeros.

1, -4, -6, -8, -9, -12, -15, -16, -25, -29, 35, -43, 51, -72, 73, -88, 96, -113, -132, -134, -154, 240, -258, -278, -368, 432, 550, -683, -847, 1060, -1078, 1240, -1447, -2573, 2845, -3448, -4610, -6226, 6695, -8565, 9132, 10246, -15335, -17334, 21777, -25071

Offset: 1

Author

M. F. Hasler, May 08 2018

Keywords

• sign
• hard

Comments

Values of A002321 at the indices listed in A304241.

These are those records of the absolute value of A002321 which are the maxima or minima between subsequent zeros. Figuratively speaking, these are the increasingly larger heights of the mountains or depths of the valleys of the graph of A002321.

Links

• Bert Dobbelaere, Table of n, a(n) for n = 1..65
• Bert Dobbelaere, C++ program for A304241 and A304242
• G. Hurst, Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture, arXiv:1610.08551 [math.NT], 2016-2017.

Crossrefs

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

Cf. A304241, A304239, A304240.

Programs

PARI

L=M=0; for(n=1,oo, if(m=merten(n), abs(m)>abs(M) && [M,N]=[m,n], abs(M)>abs(L) && (L=M) && print1(L","); M=0))

PARI

print1(j=1);for(i=1,#A051402-1,while( A028442[j] < A051402[i], j++); if( A028442[j-(j>1)]<=A051402[i] && A028442[j] < A051402[i+1], print1(","A002321(A051402[i])))) \\ Using precomputed vectors A002321 and A051402, e.g. from the b-files: {c=0;AX=apply(t->fromdigits(digits(t)[#Str(c++)+1..-1]),readvec("/tmp/bX.txt"))}

Formula

a(n) = A002321(A304241(n)).

Extensions

More terms from Bert Dobbelaere, Oct 30 2018