Sums of Powers of Primes in Arithmetic Progression

Gerard and Washington proved that, for $k>-1$, the number of primes less than $x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes up to $x$. We extend this result to primes in arithmetic progressions: we prove that the number of primes $p\equiv n(\mathrm{m}\mathrm{o}\mathrm{d}m)$ less than $x^{k+1}$ is asymptotic to the sum of $k$-th powers of all primes $p\equiv n(\mathrm{m}\mathrm{o}\mathrm{d}m)$ up to $x$. We prove that the prime power sum approximation tends to be an underestimate for positive $k$ and an overestimate for negative $k$, and quantify for different values of $k$ how well the approximation works for $x$ between ${10}^4$ and ${10}^8\mathrm{.}$