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

2011/03/01 by Lamzouri, Youness
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1103.0060

Abstract

Let q≥ 3, 2≤ r≤ ϕ(q) and a₁,...,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;a1,...,ar(x)=(E(x;q,a₁),..., E(x;q,a_r)), where E(x;q,a)= (log x)/(√(x))(ϕ(q)π(x;q,a)-π(x)), has a limiting distribution μ_q;a1,...,ar which is absolutely continuous on ℝ^r. Under the same assumptions, we determine the asymptotic behavior of the large deviations μ_q;a1,...,ar(||\vx||>V) for different ranges of V, uniformly as q→∞.

Related

• Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula\n for the densities
• Prime number races with three or more competitors
• Biases in the Shanks-Renyi Prime Number Race
• The Shanks-Rényi prime number race with many contestants
• Inequities in the Shanks-Renyi prime number race over function fields