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Sums of Powers of Primes in Arithmetic Progression

Muhammet Boran, John Byun, Zhangze Li, Steven J. Miller, Stephanie Reyes

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 $10^8.$

Sums of Powers of Primes in Arithmetic Progression

Abstract

Gerard and Washington proved that, for , the number of primes less than can be well approximated by summing the -th powers of all primes up to . We extend this result to primes in arithmetic progressions: we prove that the number of primes less than is asymptotic to the sum of -th powers of all primes up to . We prove that the prime power sum approximation tends to be an underestimate for positive and an overestimate for negative , and quantify for different values of how well the approximation works for between and
Paper Structure (22 sections, 5 theorems, 76 equations, 16 tables)

This paper contains 22 sections, 5 theorems, 76 equations, 16 tables.

Table of Contents

  1. Introduction
  2. Historical Context
  3. Statement of Main Results
  4. Proofs of Theorems
  5. Proof of Theorem \ref{['1:thm']}
  6. Start of Proof
  7. Error Term
  8. Conclusion
  9. Proof of Theorem \ref{['thm:error-pi-mn']}
  10. Abel's Summation Formula and an Additional Lemma
  11. Proof of Theorem \ref{['thm:error-pi-mn']} by Lemma \ref{['lem:error-pi-mn']}
  12. Appendices
  13. Appendix: Some Examples of Main Result
  14. Error for $k=1$
  15. Error for $k=1/2$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Fix a real number $k>-1$ and positive integers $m,n\in\mathbb{Z}_{>0}$ such that $\gcd(m,n)=1$. Then we can approximate the number of primes $p\equiv n\pmod{m}$ less than $x^{k+1}$ by the sum of $k$-powers of primes $p\equiv n\pmod{m}$ less than a real number $x$: where $\alpha$ is a positive constant.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • proof
  • ...and 5 more