Timeline for Converse to a theorem of Landau on Dirichlet series
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 31, 2011 at 10:55 | vote | accept | Kevin Smith | ||
| Oct 31, 2011 at 7:37 | comment | added | Kevin Smith | $M(n)$ is Mertens function (the last term counted with weight $1/2$). | |
| Oct 31, 2011 at 0:20 | comment | added | Frank Thorne | Sorry, what is $M(n)$? | |
| Oct 30, 2011 at 23:41 | answer | added | Greg Martin | timeline score: 3 | |
| Oct 30, 2011 at 23:34 | comment | added | Kevin Smith | One can see that it is not of constant because $M(x)$ is known to be zero infinitely often. | |
| Oct 30, 2011 at 23:23 | comment | added | Kevin Smith | Error: should read $c_n=2\mu(n)M(n)$. | |
| Oct 30, 2011 at 23:21 | comment | added | Kevin Smith | Thanks for the input, Frank. I am interested in the sequence $c_n=2\mu{n}M(n)$- its partial sum is $M^2(x)$ - and so satisfies the statement I made in the question. I don't expect it to be of constant sign for sufficiently large $n$ (that would be asking a lot!), yet I am wondering if the pole on the real axis (which occurs at $2\Omega$, where $\Omega$ is the largest real part of a zero of $\zeta(s)$) + some other hypotheses would imply the positivity of the partial sum. | |
| Oct 30, 2011 at 20:24 | comment | added | Frank Thorne | As Daniel explained, you can rig counterexamples to any simple statement you might hope to prove. However, theorems of this flavor tend to be provable under conditions on the $c_n$ which are strong, but typical for the kinds of Dirichlet series that show up in number theory. What example are you interested in? | |
| Oct 30, 2011 at 19:33 | answer | added | Daniel Loughran | timeline score: 4 | |
| Oct 30, 2011 at 18:57 | history | edited | Kevin Smith | CC BY-SA 3.0 |
added 45 characters in body; deleted 45 characters in body
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| Oct 30, 2011 at 18:51 | history | asked | Kevin Smith | CC BY-SA 3.0 |