The well known partition function $p(n)$ is defined as the number of ways to represent $n$ as the sum of natural numbers. An asymptotic formula for $p(n)$ is
$$p(n)\sim\frac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right)$$
which was obtained by Ramanujan. Recently an interesting idea came to me: generalizing the partition function. The number of ways of representing $n$ as the sum of four squares is known, and many similar things like number of ways of representing a number as a sum of two squares, etc. are known. But that didn't satisfy me. I wanted to truly generalize it.

So first, I took $p_2(n)$, the number of ways of representing $n$ as the sum of squares. It is obvious that $p_2(n)\le p(n)$. It has been conjectured that
$$p_2(n)\sim c\cdot n^{\alpha}\exp(\beta\cdot n^{1/3})$$
where

- $\alpha=-\frac{7}{6}$
- $\beta=\frac{3}{2}\frac{\pi}{2}^{1/3}\zeta\left(\frac{3}{2}\right)^{2/3}$
- $c=\frac{\zeta(3/2)^{2/3}}{\sqrt{3}(4\pi)^{7/6}}$ and the generating function of $p_2(n)$ is $$\prod_{m\ge1}\frac{1}{1-n^{m^2}}$$ I found these in an article which was not at all about $p_2(n)$ but these two were given for some reason. So my main questions are:
- What more is known about $p_2(n)$?
- What is the generating function of $r_k(n)$? What is $r_k(n)$ is mentioned below.

And the questions which are not necessary to answer but they would be useful for me are:

- How was the conjecture even formulated? I don't think it was formulated because of computational evidence because the formula is too much complicated.
- How can we prove that generating function formula?

Update: $p_2(n)$ is on OEIS as entry A001156. From that page, I found that $$p_2(n)\sim3^{-1/2}(4\pi n)^{-7/6}\zeta\left(\frac{3}{2}\right)^{2/3}\exp(3\cdot2^{-4/3}\pi^{1/3}\zeta\left(\frac{3}{2}\right)^{2/3}n^{1/3})$$ Which was proven by Hardy and Ramanujan. Can anyone link an article containing the proof of this asymptotic formula?

See the paper

- G. H. Hardy, S. Ramanujan,
*Asymptotic formulæ in combinatory analysis*, Proceedings of the London Mathematical Society (series 2)**17**(1918) pp75—115, doi:10.1112/plms/s2-17.1.75, (scanned pdf, retypeset pdf).

$p_2(n)$ is not a standard notation; but a standard notation if $r_k(n)$ (many authors use it), which denotes the number of ways of representing $n$ as a sum of $k$ squares.

Proceedings of National Academy of Sciences, (Washington), Vol. IV, 1918, pp. 189 – 193, and later (in fuller form and with a slightly different title) inTransactions of American Mathematical Society, Vol. XXI, 1920, pp. 255 – 284. $\endgroup$ – Paramanand Singh Nov 17 '20 at 14:50On the Representation of a Number as the Sum of Any Number of Squares, and in Particular of Five or Seven(doi.org/10.1073/pnas.4.7.189) and the one in TAMS is titledOn the representation of a number as the sum of any number of squares, and in particular of five(doi.org/10.1090/S0002-9947-1920-1501144-7) $\endgroup$ – theHigherGeometer Nov 19 '20 at 5:09