Timeline for Asymptotic for Ramanujan's $\tau$-function
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2023 at 0:42 | comment | added | Will Sawin | The idea of studying the lim sup is interesting (no idea if it's been done in the literature before) but, as GH from MO suggests, certainly it's not possible to do with this formula. Instead one can use the multiplicativity - I think it might be possible to prove from this that the lim sup is something like $n^{11/2} e^{ (c+o(1))\log n/\log \log n}$ for an explicit constant $c$ (and the lim inf is the same with a minus sign in front). | |
| Nov 7, 2023 at 23:59 | comment | added | GH from MO | Note that identities like the one in your post are pretty useless for analytic purposes. In particular, they do not reveal that $\tau(p)/p^{11/2}$ for prime values of $p$ fluctuates between $-2$ and $+2$. They also do not reveal that $\tau(n)$ is a multiplicative function and in fact satisfies the Hecke multiplicativity relation. | |
| Nov 7, 2023 at 23:56 | history | answered | Ethan Splaver | CC BY-SA 4.0 |