 :I just did some calculations.  Let P(N) be the proportion of 0's in the concatenation of the integers 1 to N inclusive.  Then P(9) = 0, P(99) = 0.047619047619, P(999) = 0.0654205607477, P(9,999) = 0.0742883591761, P(99,999) = 0.0795456637396, P(999,999) = 0.0830188852261, P(9,999,999) = 0.0854838724428.   