TY - JOUR
AU - Muhammet Boran
AU - John Byun
AU - Zhangze Li
AU - Steven J. Miller
AU - Stephanie Reyes
PY - 2024/01/31
Y2 - 2024/06/21
TI - Sums of Powers of Primes in Arithmetic Progression
JF - The PUMP Journal of Undergraduate Research
JA - PUMP J. Undergrad. Res.
VL - 7
IS - 0
SE - Articles
DO - 10.46787/pump.v7i0.3918
UR - https://journals.calstate.edu/pump/article/view/3918
AB - 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.
ER -