What is the most precise upper bound for the partitions funcion $p(n)$?

Question

In the paper "Asymptotic formulæ in combinatory analysis, 1918" Hardy and Ramanujan gave an upper bound $p(n) < \frac{K}{n}e^{2\sqrt{2n}}, K > 0$. Is this the best upper bound? What is $K$'s value?

Answer 1

Erdös proved the upper bound $$ p(n)\le e^{\pi \sqrt{2n/3}}, $$ and Radmacher's formula for the partition function $p(n)$ is given by

$$p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}A_{k}(n)\sqrt{k}\left[\frac{d}{dx}\frac{\sinh\left(\frac{\pi}{k}\sqrt{\frac{2}{3}\left(x-\frac{1}{24}\right)}\right)}{\sqrt{x-\frac{1}{24}}}\right]_{x=n}$$ where $$A_{k}(n)=\sum_{h=0, (h,k)=1}^{k-1}e^{\pi i(s(h,k)-2n\frac{h}{k})}$$ and $$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\left\lfloor\frac{hr}{k}\right\rfloor-\frac{1}{2}\right)$$

Here values of $p(n)$ can be obtained by rounding sufficiently accurate truncations of Rademacher's formula.