On the prime values represented by polynomials
Many conjectures have been made concerning the infinitude of prime values assumed by the irreducible polynomials in Z[x]. The most general thus far is the Bateman-Horn conjecture. While the Bateman-Horn conjecture remains open, “on average” results have been given by Baier and Zhao [BZ2], [BZ4] for...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Theses and Dissertations |
| Language: | English |
| Published: |
2013
|
| Subjects: | |
| Online Access: | http://hdl.handle.net/10356/53714 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | Many conjectures have been made concerning the infinitude of prime values assumed by the irreducible polynomials in Z[x]. The most general thus far is the Bateman-Horn conjecture. While the Bateman-Horn conjecture remains open, “on average” results have been given by Baier and Zhao [BZ2], [BZ4] for quadratic polynomials. The Hardy-Littlewood circle method is the primary tool used in [BZ2]. Here, we use the circle method to extend [BZ2] to cubic polynomials. To this end, we need to use the large sieve for cubic Dirichlet characters due to Baier and Young [BY], a large sieve for algebraic number fields due to M.N. Huxley [H], Artin reciprocity, and bounds for exponential sums. We also discuss what might be needed for higher degree polynomials. Furthermore, we discuss an application of the Bateman-Horn conjecture to the density of suitably normalized polynomial roots to prime moduli in the unit interval. |
|---|