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We generalize current known distribution results on Shanks--R\'enyi primenumber races to the case where arbitrarily many residue classes are involved.Our method handles both the classical case that goes back to Chebyshev andfunction field analogues developed in the recent years. More precisely, let$\pi(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo$q$. For a fixed integer $q$ and distinct invertible congruence classes$a_0,a_1,\ldots,a_D$, assuming the generalized Riemann Hypothesis and a weakversion of the linear independence hypothesis, we show that the set of $x$ realfor which the inequalities $\pi(x;q,a_0)>\pi(x;q,a_1)> \ldots >\pi(x;q,a_D)$are simultaneously satisfied admits a logarithmic density.

Limiting properties of the distribution of primes in an arbitrarily large number of residue classes