# Does Dirichlet's theorem on arithmetic progressions work with other notions of density?

Dirichlet's theorem on arithmetic progressions is that the density of the prime numbers for each residue class $a$ mod $n$ with $(a, n) = 1$ is $\varphi(n)^{-1}$. I believe this is easier to prove with Dirichlet density, however it also works with natural density due to the much stronger Chebotarev's density theorem.

My question is, does this theorem hold for other notions of density? In particular, I am interested in the following notion: let $P_{a, n}$ be the set of prime numbers in the arithmetic progression $a+nd$ with $(a, n) = 1$. Let $P_{a, n, m} = P_{a, n} \cap \{1, 2, \dots, m\}$. What can be said about the density:

$$\lim_{m \to \infty} \frac{\sum_{x \in P_{a, n, m}}x}{m(m+1)/2}?$$

• it works as for $\sum_{p < x} p$. this is because what we can prove for $\zeta(s)$ and $\log \zeta(s) = \sum_{p^k} \frac{(p^k)^{-s}}{k}$ we also know how to prove it for $L(s,\chi)=\sum_n \chi(n) n^{-s}$ and $\log L(s,\chi) = \sum_{p^k} \chi(p^k)\frac{(p^k)^{-s}}{k}$, knowing that $\sum_{p^k \equiv a \bmod q} \frac{(p^k)^{-s}}{k} = \frac{1}{\varphi(q)} \sum_{\chi \bmod q} \chi(a) \log L(s,\chi)$ (the Dirichlet characters work exactly as the discrete Fourier transform on $1\ldots q-1, gcd(a, q) = 1$ ) – reuns Jul 13 '16 at 19:27
• so overall $\sum_{p < x} p \sim \sum_{n < x} \frac{n}{\ln n} \sim\frac{1}{2} \frac{x^2}{ \ln x}$ and $\sum_{p < x, p \equiv a \bmod q} p \sim \frac{1}{\varphi(q)} \sum_{p < x} p \sim \frac{1}{2 \varphi(q)}\frac{x^2}{\ln x}$ – reuns Jul 13 '16 at 19:37