Annoying, boring time waster problem with probably no good solution.

Question

kainui · Answer

Evaluate $$\sigma(n)^{\tau(n)} \mod n$$

$\sigma(n)$ is the sum of divisors of n and $\tau(n)$ is the number of divisors of n.

brainzonly · Answer

???? are those convolution sums?

brainzonly · Answer

this is definitely a waste of valuable brain cells. I think I've wasted plenty already lol

kainui · Answer

Yeah, they can be represented as convolutions but this isn't really haha.

brainzonly · Answer

https://en.wikipedia.org/wiki/Divisor_function or https://www.researchgate.net/publication/274184131_Evaluation_of_the_convolution_sums_l36mn_slsm_and_4l9mn_slsm

brainzonly · Answer

I seriously don't know

kainui · Answer

Not really sure why I thought this might be interesting, but since you have all the divisors added, when you raise it to the power of the number of divisor you're bound to get terms that are exactly equal to $n$ itself based on the number of ways it combines, for instance,

$$(1+2+3+6)^4$$ contains the terms $1*2*3$ several times which will get modded out.

brainzonly · Answer

that's what I was thinking too but wasn't really sure what it had to do with the problem

kainui · Answer

That paper's kind of interesting, defining these $W_{a,b}(n)$ sums; do you know if they can be represented in terms of the Dirichlet convolution? Or what series actually makes them convolutions?

brainzonly · Answer

my internet's shotty, so if I don't reply don't worry

kainui · Answer

What is this stuff, this reminds me of some modular forms stuff but I have not dug too deep into it to know but it looks fun. Is there any particular motivation for why anyone would be interested in all these sums?

brainzonly · Answer

no I don't know what makes them convolutions, they just reminded me of them..

kainui · Answer

I'm more interested in learning something new if you got it rather than throw my self at some random garbage I made up, if you have any interesting math stuff you wanna share. What sorta math stuff are you into?

brainzonly · Answer

I don't know what the motivation is. It sounds complicated!!! :)

brainzonly · Answer

everything from basics to calculus & trig. :)

kainui · Answer

Well are you familiar with the Dirichlet convolution or the Riemann zeta function or not really into that sorta stuff yet haha

vuriffy · Answer

1 + 1 = 2.

vuriffy · Answer

My life motto. That's what I am familiar with. (:

brainzonly · Answer

no I kinda forgot. the process is complicated!!!!