How to approximate the sums of powers of primes is an interesting topic. Previous research 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. I extend this result to primes in arithmetic progressions: I prove that the number of primes p=n mod m less than x^{k+1} is asymptotic to the sum of k-th powers of all primes p= n mod m up to x. I also 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. In this seminar, I will talk about the history of the prime number theorem and my findings.