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 $$ \frac{x^{1+\alpha}}{(1+\alpha)\phi(l)\log x}. $$

$\begingroup$ 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 $a$? $\endgroup$ – Iguana Oct 28 '19 at 6:35