Timeline for On the proof of Theorem 15.2 in Montgomery-Vaughan's Multiplicative number theory
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2020 at 13:10 | vote | accept | gp1 | ||
| Mar 10, 2020 at 4:00 | history | became hot network question | |||
| Mar 9, 2020 at 21:46 | comment | added | GH from MO | I explained in the previous remark what's going on. Bailleul explained the same in more detail (see his response). The point is to consider real values $s$ initially, then half-planes afterwards. | |
| Mar 9, 2020 at 21:46 | comment | added | Wojowu | @GHfromMO I see, thanks. I didn't quite realize Lemma 15.1 is crucial for this part of the argument. | |
| Mar 9, 2020 at 21:43 | comment | added | GH from MO | @Wojowu: Initially, the LHS is only analytic in a neighborhood of the real interval $(\Theta-\varepsilon,\infty)$. So the RHS has analytic continuation to a neighborhood of the real interval $(\Theta-\varepsilon,\infty)$, hence by Lemma 15.1, also to the half-plane $\Re(s)>\Theta-\varepsilon$. It follows that the LHS is analytic in the half-plane $\Re(s)>\Theta-\varepsilon$, which contradicts the definition of $\Theta$. | |
| Mar 9, 2020 at 21:21 | answer | added | A. Bailleul | timeline score: 8 | |
| Mar 9, 2020 at 20:08 | comment | added | gp1 | @Wojowu, yes. But as you can see in that proof, the initial equation was defined for $\Re(s)>1$, but then was extended to $\Re(s) > \Theta - \epsilon$. It seems they did this by applying the identity theorem for holomorphic functions (?), which requires both sides of the equation to be analytic for $\Re(s) > \Theta - \epsilon$ | |
| Mar 9, 2020 at 20:01 | comment | added | Wojowu | Consider the displayed equation just below (15.3). The LHS is clearly analytic for $\Re(s)>\Theta-\varepsilon$. | |
| Mar 9, 2020 at 20:00 | review | First posts | |||
| Mar 9, 2020 at 20:09 | |||||
| Mar 9, 2020 at 19:56 | history | asked | gp1 | CC BY-SA 4.0 |