Values of random polynomials under the Riemann hypothesis

The well-known result states that the square-free counting function up to N is N/ζ(2)+O(N1/2). This corresponds to the identity polynomial Id(x). It is expected that the error term in question is O_ε(N1/4+ε) for arbitraliy small ε>0. Usually, it is more difficult to obtain similar order of error term for a higher degree polynomial $f(x)$ in place of Id(x). Under the Riemann hypothesis, we show that the error term, on average in a weak sense, over polynomials of arbitrary degree, is of the expected order O_ε(N1/4+ε).