Large deviations of the limiting distribution in the Shanks-Rényi prime number race

Let $q\geq 3$, $2\leq r\leq ϕ(q)$ 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 $E(x;q,a)= \frac{\log x}{\sqrt{x}}(ϕ(q)π(x;q,a)-π(x))$, has a limiting distribution $μ_{q;a_1,...,a_r}$ which is absolutely continuous on $\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $μ_{q;a_1,...,a_r}(||\vx||>V)$ for different ranges of $V$, uniformly as $q\to\infty.$

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