You don't owe anything in federal income taxes. Federal income taxes were declared unconstitutional by the supreme court in 1895 and the decision stood until the 16th amendment in 1913 when they were declared constitutional and mandatory again.
During those years, the US didn't receive any money at all from it's citizens and because of this the US reserve started to decrease.
If you would owe them, however, they would be paid directly to the US Federal Government.