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

TL;DR

This paper investigates the large deviation probabilities of the limiting distribution related to prime number races, under certain hypotheses, as the modulus q grows large, providing asymptotic estimates for these probabilities.

Contribution

It determines the asymptotic behavior of large deviations in the prime number race distribution as q approaches infinity, under GRH and GSH assumptions.

Findings

Asymptotic estimates for large deviation probabilities are derived.

Results are uniform across different ranges of deviation V.

Analysis applies to the distribution of prime counts in residue classes.

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 $q\to\infty.$