Sums of Powers of Primes in Arithmetic Progression

  • Muhammet Boran Yıldız Technical University
  • John Byun Carleton College
  • Zhangze Li University of Michigan
  • Steven J. Miller Williams College
  • Stephanie Reyes Claremont Graduate University
Keywords: prime number theorem; arithmetic progression


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.

How to Cite
Boran, M., Byun, J., Li, Z., Miller, S. J., & Reyes, S. (2024). Sums of Powers of Primes in Arithmetic Progression. The PUMP Journal of Undergraduate Research, 7, 29-50.