2011/08/26 by Lamzouri, Youness
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1108.5342
Under certain plausible assumptions, M. Rubinstein and P. Sarnak solved the Shanks--Rényi race problem, by showing that the set of real numbers x≥ 2 such that π(x;q,a1)>π(x;q,a2)>...>π(x;q,ar) has a positive logarithmic density δq;a1,...,ar. Furthermore, they established that if r is fixed, δq;a1,...,ar→ 1/r! as q→ ∞. In this paper, we investigate the size of these densities when the number of contestants r tends to infinity with q. In particular, we deduce a strong form of a recent conjecture of A. Feuerverger and G. Martin which states that δq;a1,...,ar=o(1) in this case. Among our results, we prove that δq;a1,...,ar∼ 1/r! in the region r=o(√(log q)) as q→∞. We also bound the order of magnitude of these densities beyond this range of r. For example, we show that when log q≤ r≤ ϕ(q), δq;a1,...,ar≪ε q-1+ε.