Independence of random products mod $p$

Question

Let $p$ be a large prime and $1 \leq q < p$ be chosen uniformly at random. Let $0 \leq r_1 \neq r_2 < p$ be arbitrary but fixed.

Question: for sufficiently large $p$, are $qr_1 \bmod p$ and $qr_2 \bmod p$ statistically independent? (i.e. if I take $p$ sufficiently large, can I make the covariance arbitrarily small?)

Answer 1

I have two questions to you:

1. Why do you allow $r_1=0$ or $r_2=0$? I think you should not.
2. Do you mean that $r_1$ and $r_2$ remain fixed when you change (said, increase) $p$?

Anyway... For $0 < r_1, r_2 < p$, the expected value of each of these two random variables is $\dfrac{p}{2}$, the variance is $\dfrac{p(p-2)}{12}$, and their covariance is $\dfrac{p^2}{p-1}D(r_1,r_2;p)$, where $D$ is the Dedekind sum; the correlation coefficient is then $\dfrac{12pD(r_1,r_2;p)}{(p-1)(p-2)}$. In particular, if $r_1$ and $r_2$ remain fixed while $p$ grows, the Rademacher identity implies that the correlation coefficient approaches nonzero limit $\gcd^2(r_1,r_2)/(r_1 r_2)$ (funny - this is GCD/LCM).