3
$\begingroup$

I am trying to find a closed form for the following sum:

$$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$

where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors.

Let's denote this $\sigma_1 \Delta \sigma_2$, and more generally $\sigma_k \Delta \sigma_l$ for the convolution of $\sigma_k$ and $\sigma_l$.

It seems that for when $k$ and $l$ are both odd, there are closed forms (linear combinations of $\sigma_i$'s usually) for $\sigma_k \Delta \sigma_l$ that come about by using Eisenstein series.

Are there closed forms for discrete convolutions $\sigma_k \Delta \sigma_l$ but when $k$ and $l$ are either both even, or of different parity?

Improve this question
$\endgroup$
6
  • 2
    $\begingroup$ @GH I think that the OP may want to express the convolution as a $\mathbb Q$-linear combination of $\sigma_i$'s, and that's what "closed form" means. $\endgroup$
    – Joe Silverman
    Commented Feb 25, 2018 at 3:55
  • $\begingroup$ You could try calculating the first few values of $\sigma_1\Delta\sigma_2$ and looking up the result in the Online Encyclopedia of Integer Sequences. $\endgroup$
    – Gerry Myerson
    Commented Feb 25, 2018 at 4:53
  • $\begingroup$ @GerryMyerson it doesn't seem to be on OEIS already. $\endgroup$
    – metallicmural99
    Commented Feb 25, 2018 at 15:38
  • $\begingroup$ @GHfromMO Joe Silverman is right. I am thinking of actually linear combinations of the $\sigma_i$ where the coefficients are polynomials in $n$. For instance $$\sigma_1 \Delta \sigma_1 = \frac{5}{12}\sigma_3(n) +( \frac{1}{12} - \frac{1}{2} n) \sigma_1(n)$$ by one of Ramanujan's formulae. $\endgroup$
    – metallicmural99
    Commented Feb 25, 2018 at 15:42
  • 1
    $\begingroup$ Note that if $k$ and $l$ are odd but not very small (e.g. both are at least $11$), you will not get such a formula, because the cuspidal spectrum also enters. $\endgroup$
    – GH from MO
    Commented Feb 26, 2018 at 0:05

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .