Timeline for On the prime number theorem in arithmetic progression
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 24, 2013 at 20:03 | vote | accept | Marco Cantarini | ||
| Oct 23, 2013 at 7:47 | comment | added | GH from MO | @Houfei: The problem is that $\beta$ is not fixed, but depends on $q$, so the lower bound you state is much harder to achieve. Linnik managed to do this for $x>q^L$ and $L$ large (the Linnik constant), and research since has focused on lowering the value of $L$. Currently we have $L=5.2$, while the Generalized Density Hypothesis (a consequence of GRH) would allow any $L>2$. | |
| Oct 23, 2013 at 2:03 | comment | added | H.Flip | $(3.3)$ in Andrew Granville paper in my comment implies when $q$ is small(perhaps $x>q^{1+\epsilon}$ can satisfies) $$\pi(x,q;a)\sim \frac{x-x^\beta}{\varphi(q)\log x}\ge (1-\epsilon) \frac{x}{\varphi(q)\log x}$$ where $\beta$ is real zero of $L(s,\chi) $ that is close to $1 $ with the assumption $\chi(a)=1 $ which can be omitted. | |
| Oct 21, 2013 at 14:39 | history | answered | GH from MO | CC BY-SA 3.0 |