Timeline for Mertens formulas aren't enough for prime number theorem
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 11, 2020 at 18:07 | comment | added | user627482 | I've another question...do you have any idea about what kind of "$\Lambda$" function does this sequence give? I mean, for the primes we have the von Mangoldt function $\Lambda(n)=\log p$ for $n=p^{k}$, and the Mertens' formula $\sum_{n\geq1}\Lambda(n)/n=\log x+O(1)$, with the sequence you have found what do we get? | |
| Nov 8, 2020 at 15:00 | comment | added | user627482 | Thanks you a lot!! | |
| Nov 8, 2020 at 14:49 | comment | added | reuns | The derivative of $\frac{x}{\ln x}(1+\frac14 \cos(\ln x))$ is $\ge 0$ and $\to 0$. ie. $a_n = \lfloor \frac{n}{\ln n}(1+\frac14 \cos(\ln n))\rfloor - \lfloor \frac{n-1}{\ln( n-1)}(1+\frac14 \cos(\ln (n-1)))\rfloor$ | |
| Nov 8, 2020 at 14:25 | comment | added | user627482 | Thanks for the answer anyway, you nailed it! But it's not clear to me why such a sequence exists. | |
| Nov 7, 2020 at 19:45 | history | edited | reuns | CC BY-SA 4.0 |
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| Nov 7, 2020 at 19:44 | comment | added | user627482 | In the first sum, did you mean $n\leq x$? | |
| Nov 7, 2020 at 19:42 | history | answered | reuns | CC BY-SA 4.0 |