Values of random polynomials under the Riemann hypothesis

The well-known result states that the square-free counting function up to N is N/ (2) +O (N^1/2). This corresponds to the identity polynomial Id (x). It is expected that the error term in question is O_ (N^1{4+}) for arbitraliy small >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_ (N^1{4+}).