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

Abstract

AbstractLet q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where $E(x;q,a)= ({\log x}/{\sqrt{x}})(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on $\mathbb{R}^r$. Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.