# Did Hardy and Ramanujan miscalculate these values?

When I read Dickson's History Of The Theory Of Numbers Vol-2, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text according to the literature. In the original text, the theorem gives the asymptotic formula of partition number.

$$p(n)\sim\sum^v_{q=1}A_q\phi_q+\mathcal{o}\left(n^{-\frac{1}{4}}\right)$$

where

$$\begin{matrix} \begin{array}{ll} A_1=1&A_2=\cos(n\pi)\\ A_3=2\cos\left(\frac{2}{3}n\pi-\frac{\pi}{18}\right)& A_4=2\cos\left(\frac{1}{2}n\pi-\frac{\pi}{8}\right)\\ A_5=2\cos\left(\frac{2}{5}n\pi-\frac{\pi}{5}\right)+2\cos\left(\frac{4}{5}n\pi\right)\qquad&A_6=2\cos\left(\frac{1}{3}n\pi-\frac{5\pi}{18}\right)\\ \end{array}\\ \begin{array}{l} A_7=2\cos\left(\frac{2}{7}n\pi-\frac{5\pi}{14}\right)+2\cos\left(\frac{4}{7}n\pi-\frac{\pi}{14}\right)+2\cos\left(\frac{6}{7}n\pi+\frac{\pi}{14}\right)\qquad\\ A_8=2\cos\left(\frac{1}{4}n\pi-\frac{7\pi}{16}\right)+2\cos\left(\frac{3}{4}n\pi-\frac{\pi}{16}\right)\\ \end{array} \end{matrix}$$

\begin{align*} \phi_q&=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{C\lambda_n}{q}\right)}{\lambda_n}\right)\\ &=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\pi\sqrt{\frac{2}{3}}\cdot\frac{1}{q}\cdot\sqrt{\frac{1}{n}-\frac{1}{24}}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\\ &=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{\pi}{q}\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\\ \end{align*}

In the original text, the first six terms of series were used to calculate p(100), and the first eight terms of series were used to calculate p(200). But Wikipedia[2] says the first five terms of the series are used to calculate p(200). However, in the process of calculating p(200), are some values marked in red incorrect?

\begin{align*} A_1\phi_1(200)&=\dfrac{1}{2\pi\sqrt{2}}\left.\left\{\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\pi\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\right\}\right|_{n=200}\\ &\approx3972998993185.896\\ A_2\phi_2(200)&=\dfrac{1}{2\pi}\left.\left\{\cos\left(n\pi\right)\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{\pi}{2}\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\right\}\right|_{n=200}\\ &\approx36282.978\\ \end{align*}

\begin{align*} &&A_3\phi_3(200)&\approx-87.584&&A_4\phi_4(200)\approx5.147&&A_5\phi_5(200)\approx1.424\\ &&A_6\phi_6(200)&\approx0.071&&A_7\phi_7(200)=0&&A_8\phi_8(200)\approx0.044\\ \end{align*}

\begin{align*} \text{My Vertical Calculation}&\qquad&\text{Original Vertical Calculation}\\ \begin{array}{r} 3972998993185.896\\ +\,\,36282.978\\ -\,\,87.584\\ +\,\,5.147\\ +\,\,1.424\\ +\,\,0.071\\ +\,0\qquad\\ +\,\,0.044\\ \hline 3972999029387.975 \end{array}&\qquad& \begin{array}{r} 3972998993185.896\\ +\,\,36282.978\\ -\,\,{\color{red}{87.\boxed{555}}}\\ +\,\,5.147\\ +\,\,1.424\\ +\,\,0.071\\ +\,0\qquad\\ +\,\,{\color{red}{0.\boxed{043}}}\\ \hline 3972999029388.004 \end{array} \end{align*}

Finally, I searched the results (3972999029387.975) in Google and found the lecture notes [5].

[1] History Of The Theory Of Numbers Vol-2 (Leonard Eugene Dickson)

https://archive.org/details/HistoryOfTheTheoryOfNumbersVolII/page/n187/mode/2up

[2] Partition function [Wikipedia]

https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Approximation_formulas

[3] Asymptotic Formulæ In Combinatory Analysis (G. H. Hardy & S. Ramanujan) [Original edition]

https://archive.org/details/hardy-ramanujan-1918-asymptotic-formulaae-in-combinatory-analysis/page/84/mode/2up

[4] Asymptotic Formulæ In Combinatory Analysis (G. H. Hardy & S. Ramanujan) [Reset edition BY LaTeX]

http://ramanujan.sirinudi.org/Volumes/published/ram36.pdf

[5] Lectures on Integer Partitions (Herbert S. Wilf)

https://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf

• Can you please summarize 2 things: what values are presented in all those references, and how did you calculate your values? In particular, did you take care to use an Extended Precision library such as gmp to ensure no floating-point precision errors occurred? – Carl Witthoft Aug 31 '20 at 11:52