@article{Boran_Byun_Li_Miller_Reyes_2024, title={Sums of Powers of Primes in Arithmetic Progression}, volume={7}, url={https://journals.calstate.edu/pump/article/view/3918}, DOI={10.46787/pump.v7i0.3918}, abstractNote={<p>Gerard and Washington proved that, for <em>k</em> > -1, the number of primes less than <em>x<sup>k</sup></em><sup>+1</sup> can be well approximated by summing the <em>k</em><sup>th</sup> powers of all primes up to <em>x</em>. We extend this result to primes in arithmetic progressions: we prove that the number of primes <em>p </em>congruent to <em>n</em> modulo <em>m</em> less than <em>x<sup>k</sup></em><sup>+1</sup> is asymptotic to the sum of <em>k</em><sup>th</sup> powers of all primes <em>p </em>congruent to <em>n</em> modulo <em>m</em> up to <em>x</em>. We prove that the prime power sum approximation tends to be an underestimate for positive <em>k</em> and an overestimate for negative <em>k</em>, and quantify for different values of <em>k</em> how well the approximation works for <em>x</em> between 10<sup>4</sup> and 10<sup>8</sup>.</p>}, journal={The PUMP Journal of Undergraduate Research}, author={Boran, Muhammet and Byun, John and Li, Zhangze and Miller, Steven J. and Reyes, Stephanie}, year={2024}, month={Jan.}, pages={29-50} }