# Sums over primes in arithmetic progressions

Do we know anything about sums over primes in arithmetic progressions like the following: $$\sum_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$ where $$q$$ is a prime and $$\alpha > 0$$? If we consider the average over this sum: $$\frac{1}{\pi(x,q,a)} \sum_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$ where $$\pi(x,q,a)$$ is the number of primes $$\le x$$ in the residue class of $$a$$ modulo $$l$$, can we say that this average will be equal for all residue classes? Can we say anything of this sort even for specific values of $$\alpha$$?

The standard analytic proof of the prime number theorem for arithmetic progressions will work here as well, just replacing $$L(s,\chi)$$ with $$L(s-\alpha,\chi)$$; the asymptotic size of your first sum will be $$\frac{x^{1+\alpha}}{(1+\alpha)\phi(l)\log x}.$$
• Does this mean that asymptotically the "average" value should be equal across residue classes since the main term for $\pi(x,q,a)$ is also not dependent on $a$? – Iguana Oct 28 '19 at 6:35