Let q ≥ 3 and 2 ≤ r ≤ ϕ (q) be positive integers, and a_1, ..., a_r be distinct reduced
residue classes modulo q. Rubinstein and Sarnak defined δ (q; a_1, ..., a_r) to be the
logarithmic density of the set of real numbers x such that π (x; q, a_1)> π (x; q, a_2)> ⋯> π (x;
q, a_r). In this paper, we establish an asymptotic formula for δ (q; a_1, ..., a_r) when r ≥ 3 is
fixed and q is large. Several applications concerning these prime number races are then
deduced. First, comparing with a recent work of Fiorilli and Martin on the case r= 2, we show …