Timeline for Where can I find a rigorous proof of this statement in the literature? : $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 11, 2021 at 18:37 | answer | added | KConrad | timeline score: 16 | |
| Oct 11, 2021 at 18:36 | comment | added | Pace Nielsen | This also appears in Titchmarsh's book "The Theory of the Riemann Zeta-function". | |
| Oct 11, 2021 at 17:57 | comment | added | Joseph Van Name | You are right. I was not being rigorous. | |
| Oct 11, 2021 at 17:57 | review | Close votes | |||
| Oct 11, 2021 at 18:47 | |||||
| Oct 11, 2021 at 17:46 | comment | added | Wojowu | @JosephVanName This argument (together with an appropriate version of Abel's theorem for Dirichlet series) shows that if the limit exists, then it is equal to $0$. However, convergence is far from clear. Questions like this are subtle and convergence depends on distribution of zeros of zeta. For instance, convergence for all $s>1/2$ is equivalent to RH. | |
| Oct 11, 2021 at 17:42 | comment | added | Anurag Sahay | @Joseph Van Name: Unless I'm missing something, it's a lot subtler than the fact that $\zeta$ has a pole at $s=1$. You need to be able to justify the interchange of limits (otherwise, for example, the prime number theorem would be much easier than it is). | |
| Oct 11, 2021 at 17:33 | history | edited | Alex M. | CC BY-SA 4.0 |
deleted 53 characters in body
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| Oct 11, 2021 at 17:29 | comment | added | Joseph Van Name | More generally, $\sum_{k=1}^{\infty}\frac{\mu(k)}{k^{s}}=\frac{1}{\zeta(s)}$. Your result follows from the fact that $s=1$ is a pole of the Riemann zeta function. | |
| Oct 11, 2021 at 17:17 | comment | added | Wojowu | math.stackexchange.com/q/1169675/127263 | |
| Oct 11, 2021 at 16:47 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading and [tag:reference-request]; deleted "thanks"
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| Oct 11, 2021 at 16:45 | comment | added | LSpice | @MicahMilinovich, I think that's an answer! | |
| Oct 11, 2021 at 16:45 | comment | added | Micah Milinovich | Landau's PhD thesis: arxiv.org/abs/0803.3787 | |
| S Oct 11, 2021 at 16:37 | review | First questions | |||
| Oct 11, 2021 at 17:34 | |||||
| S Oct 11, 2021 at 16:37 | history | asked | Coxi | CC BY-SA 4.0 |