I was looking through some old notes of mine and I came across a couple lemmas/identities I wrote down in regards to a [question I asked about four years ago](https://mathoverflow.net/questions/159138/asymptotic-behavior-of-texta-x-b-y-in-mathbbn4-text-axby-n). In particular I wrote that for an arbitrary fixed integer $k>1$ we have the following asymptotic expansion:


$$\Psi\_k(n)=\left|\{(\mathbf{u},\mathbf{v})\in \mathbb{N}^k\times \mathbb{N}^k:\mathbf{u}\cdot\mathbf{v}=n\}\right|\sim \frac{\sigma\_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}$$


With $\sigma\_{k-1}(n)=\sum\_{d\mid n}d^{k-1}$ and $\zeta(k)=\sum\_{n=1}^{\infty}\frac{1}{n^k}$ the Riemann zeta function. Now alternatively if one expands the inner product of $\mathbf{u}\cdot\mathbf{v}$ we see $\Psi\_k$ can be expressed in a number of other ways:
$$\Psi\_k(n)=\sum\_{\substack{\mathbf{u}\cdot\mathbf{v}=n\\(\mathbf{u},\mathbf{v})\in \mathbb{N}^k\times \mathbb{N}^k}}1=\left|\{(u\_1,v\_1,\ldots u\_k,v\_k)\in \mathbb{N}^{2k}:n=u\_1v\_1+\cdots +u\_kv\_k\}\right|\\=\sum\_{\substack{m\_1+m\_2+m\_3+\cdots +m\_k=n\\(m\_1,m\_2,m\_3,\ldots ,m\_k)\in \mathbb{N}^k}}d(m\_1)d(m\_2)d(m\_3)\cdots d(m\_k)$$


Where $d(m)=\sum\_{d\mid m}1=\sigma\_{0}(m)$ counts the divisors of any natural number $m$. Which unearths a number of $q$-series esque representation for the ordinary generating function of $\Psi\_k$, for example:


$$\sum\_{n=1}^{\infty}\Psi\_k(n)q^n=\left(\sum\_{n=1}^{\infty}q^{n^2}\frac{1+q^n}{1-q^n}\right)^k=\frac{\text{log}(1-q)^k}{\text{log}(q)^k}\sum\_{j=0}^k\binom{k}{j}\left(\frac{\psi\_{q}(1)}{\text{log}(1-q)}\right)^j$$


Where $\psi\_{q}(z)=\frac{1}{\Gamma\_{q}(z)}\frac{d}{dz}\Gamma\_{q}(z)$ is the $q$-analog of the digamma function defined analogously in terms of the [$q$-gamma function](https://en.wikipedia.org/wiki/Q-gamma_function), expressible as $\Gamma\_{q}(z)=(1-q)^{1-z}\prod\_{n=0}^{\infty}\frac{1-q^{n+1}}{1-q^{n+z}}$ for $|q|<1$.




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Now working with just some of these alternate representations, as well as fiddling with the order of the summands involved I was able to prove by induction on the integer $k>1$ that we have both:


$$\sum\_{n\leq N}\Psi\_k(n)=\frac{N^k\text{log}(N)^{k}}{k!}+\mathcal{O}(N^k\text{log}(N)^{k-1})$$
$$\sum\_{n\leq N}\frac{\sigma\_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}=\frac{N^k\text{log}(N)^{k}}{k!}+\mathcal{O}(N^k\text{log}(N)^{k-1})\\$$


So using the same heuristics as before it seems reasonable that $\Psi\_k(n)\sim \frac{\sigma\_{k-1}(n)\text{log}(n)^{k}}{\zeta(k)(k-1)!}$ which at least for the case at $k=2$ would coincide with the answer to my previous question. However I'm unable to find a concrete proof of this result and would therefore appreciate any help in the matter.




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The number of representations of an integer as the inner product of integral lattice points
The asymptotic formula is true for even dimensions $k\geq 2$. We can prove this by induction on $k$, inspired by Rodrigo's observation on Eisenstein series. 


The case $k=2$ is classical and addressed in the OP's previous post that he linked. Now it suffices to show that if the formula is true for even dimensions $k,\ell\geq 2$, then it is also true for dimension $k+\ell$. Moreover, we can assume in this implication that either $k,\ell\geq 4$ or $k+\ell\in\{4,6\}$. Let us start from the obvious identity
$$ \Psi\_{k+\ell}(n)=\sum\_{r+s=n}\Psi\_k(r)\Psi\_\ell(s).$$
Using the fact that the logarithm is a slowly changing function, we are left with proving that
$$ \frac{\sigma\_{k+\ell-1}(n)}{\zeta(k+\ell)\Gamma(k+\ell)}\sim\sum\_{r+s=n}\frac{\sigma\_{k-1}(r)}{\zeta(k)\Gamma(k)}\cdot\frac{\sigma\_{\ell-1}(r)}{\zeta(\ell)\Gamma(\ell)}.$$
If $k,\ell\geq 4$, then the left hand side is $(2\pi i)^{-k-\ell}$ times the $n$-th Fourier coefficient of the standard holomorphic Eisenstein series $E\_{k+\ell}$ of weight $k+\ell$ and full level, while the right hand side is $(2\pi i)^{-k-\ell}$ times the $n$-th Fourier coefficient of $E\_kE\_\ell$. As $E\_{k+\ell}-E\_kE\_\ell$ is a cusp form, the result follows from standard (or even weaker) upper bounds for the Fourier coefficients of cusp forms. 


If $k+\ell\in\{4,6\}$, then the result follows from the explicit identities
$$5\sigma\_3(n)+(1-6n)\sigma\_1(n)=12\sum\_{r+s=n}\sigma\_1(r)\sigma\_1(s),$$
$$21\sigma\_5(n)+(10-30n)\sigma\_3(n)-\sigma\_1(n)=240\sum\_{r+s=n}\sigma\_1(r)\sigma\_3(s).$$
These identities can also be proved with the help of Eisenstein series, or by elementary means, see e.g. (3.10) and (3.12) in the [paper of Huard et al.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.531.7900&rep=rep1&type=pdf)


The proof is complete.


