Statistical Distribution of Roots of a Polynomial Modulo Primes III

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ $(a_{n-1},\dots,a_0\in\mathbb Z)$ be a polynomial with complex roots $\alpha_1,\dots,\alpha_n$ and suppose that a linear relation over $\mathbb Q$ among $1,\alpha_1,\dots,\alpha_n$ is a multiple of $\sum_i\alpha_i+a_{n-1}=0$ only. For a prime number $p$ such that $f(x)\bmod p$ has $n$ distinct integer roots $0<r_1<\dots<r_n<p$, we proposed in a previous paper a conjecture that the sequence of points $(r_1/p,\dots,r_n/p)$ is equi-distributed in some sense. In this paper, we show that it implies the equi-distribution of the sequence of $r_1/p,\dots,r_n/p$ in the ordinary sense and give the expected density of primes satisfying $r_i/p<a$ for a fixed suffix $i$ and $0<a<1$.