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:
$$ \frac{1}{\pi(x,l,a)} \sum\_{\substack{q \equiv a (\text{mod } l) \\ q \le x}} q^{\alpha}$$
where $\pi(x,l,a)$ 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 $\alpha$?



Sums over primes in arithmetic progressions
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
$$
\frac{x^{1+\alpha}}{(1+\alpha)\phi(l)\log x}.
$$


