That is called the Liouville function; you might get luckier searching for that name. I don't know much about the state-of-the-art on computing it, but I found

That is called the Liouville function; you might get luckier searching for that name. I don't know much about the state-of-the-art on computing it, but I found a description of such a method here, which may be a starting point to compare with: http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960-0120198-5/ http://www.ams.org/journals/mcom/1960-14-072/S0025-5718-1960...