Large deviations of the limiting distribution in the Shanks-R\'enyi prime number race

Abstract

Let $q\geq 3$, $2\leq r\leq \phi(q)$ and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis, M. Rubinstein and P. Sarnak showed that the vector-valued function $E_{q;a_1,...,a_r}(x)=(E(x;q,a_1),..., E(x;q,a_r)),$ where $E(x;q,a)= \frac{\log x}{\sqrt{x}}(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution $\mu_{q;a_1,...,a_r}$ which is absolutely continuous on $\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $\mu_{q;a_1,...,a_r}(||\vx||>V)$ for different ranges of $V$, uniformly as $q\to\infty.$