Sums of Powers of Primes in Arithmetic Progression

TL;DR

This paper extends a prime sum approximation method to primes in arithmetic progressions, analyzing its accuracy and behavior for various powers and ranges of x, providing quantitative estimates.

Contribution

It generalizes the prime power sum approximation to primes in arithmetic progressions and quantifies its accuracy for different powers and ranges.

Findings

Approximation tends to underestimate for positive k.

Approximation tends to overestimate for negative k.

Accuracy varies with x between 10^4 and 10^8.

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 $p\equiv n \pmod m$ less than $x^{k+1}$ is asymptotic to the sum of $k$-th powers of all primes $p\equiv n \pmod m$ 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 $10^8.$