OFFSET
1,2
COMMENTS
It was once conjectured that the sum of Liouville's function was never > 0 except for the first term.
It follows from Theorem 2 in Borwein-Ferguson-Mossinghoff that a(n) < 262*n^2 infinitely often, improving on an earlier result of Anderson & Stark. - Charles R Greathouse IV, Jun 14 2011
a(830) > 2 * 10^14 (probably around 3.511e14) and a(1160327) = 351753358289465 according to the calculations of Borwein, Ferguson, & Mossinghoff. - Charles R Greathouse IV, Jun 14 2011
3.75 * 10^14 < a(1160328) <= 23156359315279877168. - Hiroaki Yamanouchi, Oct 04 2015
From Jianing Song, Aug 06 2021: (Start)
a(n) is the smallest m such that A002819(m) = n.
REFERENCES
R. J. Anderson and H. M. Stark, Oscillation theorems, Analytic Number Theory (1980); Lecture Notes in Mathematics 899 (1981), pp. 79-106.
LINKS
Donovan Johnson and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100000 (terms a(1)-a(829) from Donovan Johnson)
P. Borwein, R. Ferguson, and M. Mossinghoff, Sign changes in sums of the Liouville function, Mathematics of Computation 77 (2008), pp. 1681-1694.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.
EXAMPLE
The sum of Liouville's function from 1 through 906150258 is 2, that is the smallest value, so a(2)=906150258.
PROG
(PARI) print1(r=1); t=0; for(n=906150257, 906400000, t+=(-1)^bigomega(n); if(t>r, r=t; print1(", "n))) \\ Charles R Greathouse IV, Jun 14 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved