Distribution of primes remainders Naively, I would expect the natural density the number of a fixed prime $p$ with remainder $m$ to the other primes to be uniform
$$
d(p,m) = \limsup\limits_{n\rightarrow\infty} \frac{a(n,m)}{n} = \frac{1}{p-1}
$$
where $A(n,m) =\{ x \equiv p \ ( \bmod \  m),   x\in\mathbb{P}_{<n} \}$, and $a(n,m)=|A(n,m)|$ and $1 \le m \le p-1$. A trivial test for small primes at $n<10^6$ seems to suggest that this may be true:
p m A(10**6, m)/n
3 1 0.499770694795
3 2 0.500216566027
5 1 0.249904456164
5 2 0.249968152055
5 3 0.250515936712
5 4 0.249598715891
7 1 0.166411883105
7 2 0.166437361461
7 3 0.166946928584
7 4 0.166488318174
7 5 0.166946928584
7 6 0.166755840913

Is there a way to show this is true for all primes and their remainders?
 A: This fact about the distribution of primes has been proved. In fact it is a consequence of Dirichlet's analytic proof of his theorem on primes in arithmetic progressions. 
What is proved is that a certain definition of "density" of sets of primes lying in a given invertible class mod $N$ is $\frac{1}{\phi(N)}$. In other words the primes are distributed uniformly amongst the classes of $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$ (except a few bad primes that divide $N$).
Your findings show this for $N$ prime.
There is a far reaching generalization of this called the Cebotarev Density theorem. The point is that the arithmetic group $\left(\mathbb{Z}/N\mathbb{Z}\right)^{\times}$ can be realised as the cyclotomic Galois group Gal$(\mathbb{Q}(\zeta_N)/\mathbb{Q})$. In fact there is a particular way to construct an isomorphism by assigning to each prime number it's Frobenius element, a very important element of the Galois group that measures arithmetic properties of splitting of primes in extensions.
Dirichlet's corollary now becomes that, for this particular extension of number fields, the prime numbers are uniformly distributed between the $[\mathbb{Q}(\zeta_N):\mathbb{Q}] = \phi(N)$ elements of the Galois group according to their corresponding Frobenius elements.
Cebotarev stated and proved the situation for any Galois extension of number fields $L/K$, except he found that you really need an abelian Galois group in order to get uniform distribution of primes amongst the $[L:K]$ possible Frobenius elements.
Why is this? Well in non-abelian extensions the Frobenius element is in general not well defined...it is really defined by a conjugacy class in the Galois group. In these cases the primes having a given Frobenius conjugacy class of size $k$ have density $\frac{k}{[L:K]}$ rather than $\frac{1}{[L:K]}$ in the abelian case.
A: Yes, this is Dirichlet's theorem on arithmetic progressions
