Demystifying the asymptotic expression for the partition function

The proof of the asymptotic formula for the partition function given by Hardy and Ramanujan was "the birth of the circle method", and used properties of modular forms. Erdös was trying to give a proof with elementary methods (he also gave a so-called elementary proof of the PNT with Selberg). He succeeded in 1942 to give such a proof, but only with "unknown" constant $a$, see here. Afterwards Newman gave a "simplified proof", and obtained also that $a=\frac{1}{4n\sqrt{3}}$, see here.

There are several "modern" references now, which give an elementary proof of the asymptotic formula for $p(n)$. Here are two references:

M. B. Nathanson: On Erdös's elementary method in the asymptotic theory of partitions.

Daniel M. Kane (was misspelled as "Cane"): An elementary derivation of the asymptotic of the partition function.

Instead of using modular forms etc. the elementary method of Erdös only uses elementary estimates of exponential sums and an induction from the identity $$ np(n)=\sum_{ka\le n}ap(n-ka). $$