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 $\alpha$?
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$\begingroup$ @MatemáticosChibchas Thanks for pointing that out. I meant to write $\pi(x,l,a)$. $\endgroup$Iguana– Iguana2022-09-04 02:46:27 +00:00Commented Sep 4, 2022 at 2:46
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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 $$ \frac{x^{1+\alpha}}{(1+\alpha)\phi(l)\log x}. $$
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$\begingroup$ 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– Iguana2019-10-28 06:35:14 +00:00Commented Oct 28, 2019 at 6:35
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$\begingroup$ Thanks for your answer. Could you please refer to a specific proof of the PNT for arithmetic progressions that can be tweaked in the form you indicate? I tried without success. $\endgroup$Matemáticos Chibchas– Matemáticos Chibchas2022-09-03 22:26:06 +00:00Commented Sep 3, 2022 at 22:26
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$\begingroup$ If you know of a proof of PNT for arithmetic progressions that uses contour integration, almost surely that proof can be adapted to this scenario as well. $\endgroup$Greg Martin– Greg Martin2022-09-03 23:12:21 +00:00Commented Sep 3, 2022 at 23:12