Sums of Powers of Primes in Arithmetic Progression
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910315453612032 |
|---|---|
| author | Boran, Muhammet Byun, John Li, Zhangze Miller, Steven J. Reyes, Stephanie |
| author_facet | Boran, Muhammet Byun, John Li, Zhangze Miller, Steven J. Reyes, Stephanie |
| contents | 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 $10^8.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16007 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sums of Powers of Primes in Arithmetic Progression Boran, Muhammet Byun, John Li, Zhangze Miller, Steven J. Reyes, Stephanie Number Theory (Primary) 11N13, (Secondary) 11N05 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 $10^8.$ |
| title | Sums of Powers of Primes in Arithmetic Progression |
| topic | Number Theory (Primary) 11N13, (Secondary) 11N05 |
| url | https://arxiv.org/abs/2309.16007 |