Abstract
Gerard and Washington proved that, for k > -1, the number of primes less than xk+1 can be well approximated by summing the kth powers of all primes up to x. We extend this result to primes in arithmetic progressions: we prove that the number of primes p congruent to n modulo m less than xk+1 is asymptotic to the sum of kth powers of all primes p congruent to n modulo 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 104 and 108.
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