¼ The constant The constant p ¼

## 3  Ramanujan type formulae

In 1914, the extraordinary mathematician from India, Srinivasa Ramanujan published a set of 14 new formulae ([1]), one of them is famous:

 1p = 2Ö29801 ¥ å k = 0 (4k)!(k!)444k (1103+26390k)994k (*)
The convergence is geometric but very impressive, let's take a look to the first iterates:

 x1
 =
 3.141592(7300133056603139961890252155185995816071...)
 x2
 =
 3.141592653589793(8779989058263060130942166450293...)
 x3
 =
 3.14159265358979323846264(90657027588981566774804...)
 x4
 =
 3.1415926535897932384626433832795(552731599742104...)
 x5
 =
 3.141592653589793238462643383279502884197(6638181...)

In his article, Ramanujan gave very little details of how he found such formulae. To find those relations, he needed modular equations and hypergeometric functions. It's possible to find a complete demonstration of those series in ([2]). In this book the following theorem is given:

 1p = gN ¥ å k = 0 (1/4)k(1/2)k(3/4)k(k!)3 ( aN+kbN)gN2k,       N > 2

where (aN,bN,gN) are algebraic numbers whenever N is rational and (a)k is the rising factorial : (a)k = a(a+1)(a+2)...(a+k-1). The expression of (aN,bN,gN) as functions of N is rather complex... For example gN is given by :

 gN = 2gN12+gN-12

where gN is a Ramanujan's invariant.

Ramanujan's formula is given for N = 58, for which g582 = (5+Ö[29])/2 so g58 = 1/992 and a58+kb58 = 2Ö2(1103+ 26390k) ([2], [4]).

Other interesting series are found with (N = 6,10,18,22,58), all given by Ramanujan in his article. Here are the value of gN and gN for the corresponding values of N:

 g6
 =
 (1+Ö2)1/6,g6 = 1/3
 g10
 =
 (2+Ö5)1/6,g10 = 1/9
 g18
 =
 (5+2Ö6)1/6,g18 = 1/49
 g22
 =
 (1+Ö2)1/2,g22 = 1/99
 g58
 =
 ((5+ __Ö29 )/2)1/2,g58 = 1/992

In the same article, Ramanujan also gave:

 1p = ¥ å k = 0 æç è 2k k ö÷ ø 3 (42k+5)212k+4

This series adds roughly 2 digits per term. A proof was published in [2].

Much later, a few years ago [3], David and Gregory Chudnovsky found a very powerful series of this type :

 1p = 12 ¥ å k = 0 (-1)k (6k)!(3k)!(k!)3 (13591409+545140134k)6403203k+3/2

### 3.1  Record of computation with the Ramanujan like formulae :

1985 : Gosper found 17,500,000 decimal digits with formula (*).

1994 : The Chudnovsky brothers found 4,044,000,000 digits with their formula.

## References

[1]
S. Ramanujan, Modular equations and approximations to p, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372

[2]
J.M. Borwein and P.B. Borwein, Pi and the AGM - A study in Analytic Number Theory and Computational Complexity, A Wiley-Interscience Publication, New York, (1987)

[3]
D. V. Chudnovsky and G. V. Chudnovsky, Approximations and complex multiplication according to Ramanujan, in Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375-396 & p. 468-472.

[4]
P. Eymard and J. P. Lafon, Autour du nombre p, Paris, Hermann, (1999)

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