Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r\_1 < r\_2 < \dots < r\_{\varphi(m)} = m-1$ be the prime residues
(mod $m$), and let $\sigma \in {\rm Sym}(\{r\_1, \dots, r\_{\varphi(m)}\})$
be a permutation of them. Are there always infinitely many
$n \in \mathbb{N}$ such that
$\pi(m,\sigma(r\_1),n) < \pi(m,\sigma(r\_2),n) < \dots < \pi(m,\sigma(r\_{\varphi(m)}),n)$?


If the answer is *yes*, are explicit bounds $B(m)$ known such that
for every permutation $\sigma$ of the prime residues (mod $m$) there is
*at least one* such $n$ less than $B(m)$?


If the answer is *no*, for which permutations $\sigma$ are there infinitely many
such $n$, and for which are not? 



Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)
The answer is conjectured to be "yes." Further, assuming GRH and GSH [Rubinstein and Sarnak, Chebyshev's Bias, Exp. Math 1994](http://projecteuclid.org/euclid.em/1048515870) even could show results on the logarithmic densities of the resepactive sets (which in particular is non-zero). I do not know about general explicit upper bounds.


Without GRH a lot less is known. The problem you mention sometimes goes by Shanks--Rényi race problem. A very good exposition of this circle of ideas is [Prime Number Races by Granville and Martin](http://arxiv.org/abs/math/0408319); see page 22 and subsequent ones for the discussion of Rubinstein and Sarnak's results. (Added note: this is in fact the same paper Benjamin Dickman mentions in a comment I had not seen before.)


