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In 1985, Gosper used the not-yet-proven formula by Ramanujan

$$\frac{ 1 }{\pi } = \frac{2\sqrt{2}}{99^2}\cdot \sum_{n=0}^\infty \frac{(4n)!}{(n!)^4}\cdot\frac{26390 n+1103}{99^{4n}}$$

to compute $17\cdot10^6$ digits of $\pi$, at that time a new world record.

Here (https://www.cs.princeton.edu/courses/archive/fall98/cs126/refs/pi-ref.txt) it reads:

There were a few interesting things about Gosper's computation. First, when he decided to use that particular formula, there was no proof that it actually converged to pi! Ramanujan never gave the math behind his work, and the Borweins had not yet been able to prove it, because there was some very heavy math that needed to be worked through. It appears that Ramanujan simply observed the equations were converging to the 1103 in the formula, and then assumed it must actually be 1103. (Ramanujan was not known for rigor in his math, or for providing any proofs or intermediate math in his formulas.) The math of the Borwein's proof was such that after he had computed 10 million digits, and verified them against a known calculation, his computation became part of the proof. Basically it was like, if you have two integers differing by less than one, then they have to be the same integer.

Now my historical question: Who was the first to prove this formula? Was it Gosper because he added the last piece of the proof, or was it the Borweins, afterwards? And was Gosper aware of this proof when he did his computation?

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    $\begingroup$ Ok, I don't think anyone has proved this formula without the aid of mathematical computation. But Ramanujan's work in this area was nearly complete and I firmly believe that he computed 1103 by hand. The reasons for this belief are the symbolic formulas which he gave in the same paper. Those symbolic formulas could not be guessed in 1914 (or earlier) rather they were obtained using algebraic manipulation. However Ramanujan does not give the symbolic formula related with this particular series in the question and no one has discovered it so far. $\endgroup$
    – Paramanand Singh
    Sep 28, 2019 at 7:09
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    $\begingroup$ Also see my related question on MO: mathoverflow.net/q/163859/15540 $\endgroup$
    – Paramanand Singh
    Sep 28, 2019 at 7:15
  • $\begingroup$ You're right! But I'm interested in what happened 1985-1987, especially the role of Gosper. Do you know something about that? $\endgroup$
    – L. Milla
    Sep 29, 2019 at 6:25
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    $\begingroup$ @reuns: does your comment refer to my answer here or to my question on mathoverflow.net? As far as the answer here is concerned all relevant definitions are provided. $\endgroup$
    – Paramanand Singh
    Sep 30, 2019 at 8:17
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    $\begingroup$ The History of Science and Mathematics StackExchange might be better-suited to addressing this as a historical question. $\endgroup$
    – Blue
    Sep 30, 2019 at 10:21

2 Answers 2

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What follows is taken directly from Borweins' Pi and the AGM.

Let $N$ be a positive number and $q_N=e^{-\pi\sqrt{N}}$ and $$k_N=\frac{\vartheta_{2}^{2}(q_N)}{\vartheta_{3}^{2}(q_N)},k'_N=\sqrt{1-k_N^2},G_N=(2k_Nk'_N)^{-1/12},g_N=\left(\frac{2k_N}{{k'} _N^{2}}\right)^{-1/12}\tag{1}$$ where $\vartheta _2,\vartheta_3$ are theta functions of Jacobi defined by $$\vartheta_{2}(q)=\sum_{n\in\mathbb {Z}} q^{(n+(1/2))^2},\, \vartheta_{3}(q)=\sum_{n\in\mathbb {Z}} q^{n^2}\tag{2}$$ Borwein brothers define another variable $$\alpha(N) =\frac{E(k'_N)} {K(k_N)} - \frac{\pi} {4K^2(k_N)}\tag{3}$$ where $K, E$ denote standard elliptic integrals $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}},\,E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx\tag{4}$$ It is well known that $k_N, k'_N, \alpha(N) $ are algebraic when $N$ is a positive rational number. We assume $N$ to be a positive integer unless otherwise stated.

Borweins present two class of series for $1/\pi$ based on Ramanujan's ideas which are of interest here: $$\frac{1}{\pi}=\sum_{n=0}^{\infty} \dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}d_n(N) x_N^{2n+1}\tag{5}$$ where $$x_N=\frac{2} {g_N^{12}+g_N^{-12}} =\frac{4k_N{k'} _{N}^{2}}{(1+k_N^2)^2}, \\ d_n(N) =\left(\frac{\alpha(N) x_N^{-1}}{1+k_N^2}+\frac{\sqrt{N} }{4}g_N^{-12}\right) +n\sqrt{N} \left(\frac{g_N^{12}-g_N^{-12}}{2}\right) \tag{6}$$ and $$\frac{1}{\pi}=\sum_{n=0}^{\infty} (-1)^n\dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}e_n(N) y_N^{2n+1} \tag{7}$$ where $$y_N=\frac{2} {G_N^{12}-G_N^{-12}} =\frac{4k_Nk'_{N}}{1-(2k_Nk'_N)^2}, \\ e_n(N) =\left(\frac{\alpha(N) y_N^{-1}}{{k'} _{N} ^{2}-k_N^2}+\frac{\sqrt{N} }{2}k_N^2G_N^{12}\right) +n\sqrt{N} \left(\frac{G_N^{12}+G_N^{-12}}{2}\right)\tag{8}$$ The series in question is based on $(5)$ with $N=58$. Another (not so famous but equally remarkable) series given by Ramanujan using $(7)$ with $N=37$ is as follows: $$\frac{4}{\pi}=\frac{1123}{882}-\frac{22583}{882^3}\cdot\frac{1}{2}\cdot\frac{1\cdot 3}{4^2}+\frac{44043} {882^5}\cdot\frac{1\cdot 3}{2\cdot 4}\cdot\frac{1\cdot 3\cdot 5\cdot 7}{4^2\cdot 8^2}-\dots\tag{9}$$ Borweins mention that the values of $\alpha(37),\alpha(58)$ (leading to $1123$ in series $(9)$ and $1103$ in series in question) were obtained by calculating $e_0(37)$ and $d_0(58)$ to high precision.

The details of these calculations are not revealed by Borwein Brothers. But it appears that using value of $\pi$ given by Gosper and the series in question (as well as series $(9)$) one can get the values of $d_0(58),e_0(37)$ to high precision. Further some amount of computation is needed to get the minimal polynomials for $\alpha(37),\alpha(58)$. And then one can get their values in closed form as radicals.

The procedure is similar to what you have done in one of your papers dealing with evaluation of coefficients in Chudnovsky formula but the analysis is probably more complicated because the functions involved here are not like the $j$ invariant taking integer values.


To sum up, Gosper role was important in this proof on two fronts. First is the computation itself and second is that his computation brought Ramanujan's formula into limelight. It was sitting there in his paper Modular equations and approximations to $\pi$ since 1914 and no one before Gosper even looked at it.

Also as most references indicate Gosper's computation was done earlier without any knowledge of Borwein's work which got published later. So Gosper was not aware of the proof of the formula and didn't know that his computation would someday be used as a part of the proof of the series which he computed.

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  • $\begingroup$ Wow, thank you! Do you have such references, as you wrote in the last section? $\endgroup$
    – L. Milla
    Oct 1, 2019 at 5:47
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    $\begingroup$ Borwein's book is my primary reference. Apart from that this is discussed (but not with details) in Pi Unleashed by Jorg Arndt. You can also read the paper by Borwein cecm.sfu.ca/organics/papers/borwein/paper/html/node15.html (search for "it is less clear"). $\endgroup$
    – Paramanand Singh
    Oct 1, 2019 at 6:44
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    $\begingroup$ @L.Milla: My guess is that Ramanujan developed alternative theories of theta functions in great detail (some of it redeveloped in Bruce Berndt's books) and perhaps the calculations in his theories were simpler than those in approach used by Borweins. $\endgroup$
    – Paramanand Singh
    Oct 1, 2019 at 6:49
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I asked Bill Gosper, and here is his response:

There was email on this at the time, which I may be able to dig up. But, as I recall, when I began my computation, the Borwein brothers had proved that if Ramanujan's formula did not equal π, it differed from π by at least 10^-3000000, so that as my computation passed the 3000000 mark in agreement with Kanada's 16000000 digit AGM computation, it served to complete the Borwein proof. But by the time my computation reached 17000000, the Borwein's resolved their ambiguity without my empirical confirmation. Their completed proof is almost certainly in their Pi and the AGM book. Tito Piezas and the Chudnovsky brothers have presumably put the matter to rest with their climactic series based on √163, the final Heegner number.

Just to clarify, I did not crank out π as a decimal string. I resumably summed Ramanujan's series as an exact rational number on a Symbolics computer with unlimited integers, converting to decimal now and then for comparison with Kanada, but with the ultimate purpose of computing the {3,7,15,1,292,...} continued fraction, which is actually mathematically interesting, as opposed to useless decimal or binary, which is actually an encryption. Regrettably, (almost) everybody ignored my continued fraction and wasted their time computing (eventually trillions of) useless digits.

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  • $\begingroup$ That's cool. +1 Bill Gosper mentions that Borwein's resolved their ambiguity without Gosper's numerical confirmation. Perhaps they might have done more research to reduce the number of digits of $\pi$ to confirm their formula and used $\pi$ from previous computations. Anyway as my answer indicates Borwein's don't give details of the computation in their book Pi and the AGM. $\endgroup$
    – Paramanand Singh
    Apr 11, 2020 at 13:30

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