Sums of Powers of Primes in Arithmetic Progression

Abstract

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 congruent to n modulo m less than x^k+1 is asymptotic to the sum of k^th 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.