Timeline for Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2020 at 2:56 | comment | added | Vincent Granville | Define $A_s(x),B_s(x), C_s(x)$ by replacing $p$ with $p^s$ in the original definition. Then the connection with $\zeta(1)=\infty$ is more obvious. For instance $f_s(x):=(1-2^{-s}) A_s(x)B_s(x) \rightarrow 1/\zeta(s)$ as $x\rightarrow\infty$. Thus $(s-1)/f_s(x)\rightarrow (s-1)\zeta(s)$. And $\lim_{s\rightarrow 1^+} (s-1)\zeta(s) = 1$. I don't know if this offers a different way to solve the problem. | |
| Oct 25, 2020 at 2:29 | comment | added | user167505 | Should the reciprocal of $\zeta(1)$ come out from somewhere? | |
| Oct 25, 2020 at 1:54 | vote | accept | Vincent Granville | ||
| Oct 25, 2020 at 1:52 | history | edited | Vincent Granville | CC BY-SA 4.0 |
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| Oct 25, 2020 at 1:28 | comment | added | Vincent Granville | Thank you! It really helps, as I am not in the Academia. | |
| Oct 25, 2020 at 1:26 | comment | added | Anurag Sahay | Here's a reference in the literature which calculates the constants you want: ams.org/journals/proc/1971-028-02/S0002-9939-1971-0277494-X | |
| Oct 25, 2020 at 1:19 | history | edited | Vincent Granville | CC BY-SA 4.0 |
edited title
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| Oct 25, 2020 at 1:19 | review | Close votes | |||
| Oct 27, 2020 at 14:11 | |||||
| Oct 25, 2020 at 1:19 | answer | added | Random | timeline score: 11 | |
| Oct 25, 2020 at 1:17 | review | Suggested edits | |||
| Oct 25, 2020 at 1:18 | |||||
| Oct 25, 2020 at 1:17 | history | edited | Vincent Granville | CC BY-SA 4.0 |
edited title
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| Oct 25, 2020 at 1:15 | comment | added | Vincent Granville | Yes these are products. For instance, $A(x)$ is the probability for a large number $N$ to to not be divisible by any prime $p\leq x$ congruent to 3 modulo 4. | |
| Oct 25, 2020 at 1:11 | comment | added | Anurag Sahay | Did you mean to use \prod and not \sum? | |
| Oct 25, 2020 at 0:56 | history | asked | Vincent Granville | CC BY-SA 4.0 |