On the Brun-Titchmarsh theorem
The Brun–Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C&am...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Published: |
Polskiej Akademii Nauk, Instytut Matematyczny
2013
|
| Summary: | The Brun–Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq≥8 and q is sufficiently large. Moreover, we also produce a lower bound of size x/(q1/2ϕ(q)) when logx/logq≥8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem. |
|---|