Values of random polynomials under the Riemann hypothesis
Values of random polynomials under the Riemann hypothesis
The well-known result states that the square-free counting function up to $N$ is $N/\zeta(2)+O(N^{1/2})$. This corresponds to the identity polynomial $\text{Id}(x)$. It is expected that the error term in question is $O_\varepsilon(N^{\frac{1}{4}+\varepsilon})$ for arbitraliy small $\varepsilon>0$. Usually, it is more difficult to obtain similar order of error term for a …