For more than 15 years, these two revolutionized
research on Pi! After the algorithm with quadratic convergence found by Brent/Salamin in 1976, they practically then monopolized the discoveries of series.
Quadratic, cubic, quadric, nonic the speed of convergence did not stop since!

In fact, they proved some years ago, that an algorithm with speed n-ic converging
Pi exists for every integer n... But the complexity of calculations increase very
fast and it seems that the algorithm with quadric convergence represents the best
ratio complexity/speed... It was moreover used in most of records since its discovery
and notably by Kanada to calculate 51 billions of decimal recently.

You will have understood it, Borwein represents with the small group composed of Chudnovsky, Simon Plouffe, Garvan, Gosper et Bailey, the highlight of the active
research on Pi today.

As
regards the proof of these formulae, I must regrettably pass besides the great principle
of this site... Because these proofs are already transcribed on Web at the following
address: www.cecm.sfu.ca/organics/papers/garvan/paper/html/paper.html. It is necessary to recognize also that it would be useless
to copy out silly 4 or 5 summaries of demonstration without being able to complete
the intermediaries of calculation (even though principle is rather simple to understand),
because not being of the level of a mathematician...

I retranscribe so only the summary of the proof for the 3) (2-nd formula), which
is perfectly representative of the principle of demonstrations using modular equations
(to see the page dedicated to Ramanujan for the explanation
of this theorie and of the principle of the démonstration)

The techniques of this proof mainly appeal to theta
functions, to the Eta function of Dedekind and to their properties.

1.1 Introduction :

So let us put Thêta functions:

, *r>0*

Then, function from the Borwein is introduced : *rR*^{+*},

When *r->+*, *q -> 0* and .

We are going to build an infinity of series _{p} and find a relation between _{p}*(N*^{2}r) and _{p}*(r)* (As the Borwein say, it is a "nice" relation
if *N=p* !)

2.1 Construction
of series _{p}

For it, we pose *q=exp(2i)* and function of Dedekind is considered :

We remind the relation found by Euler :

2.2 This
Eta function admits as property, among further :

2.3 Let us
put besides :

2.4 And
let us put finally (yeah!) _{p} function defined by :

(Easy to remember, of course...)

2.5 We
have :

*A*_{p}(r)=1+O(q)

2.6 and
according to 2.2,

We have then

(All these previous relations are very difficult to find but fortunately true (!)
and principle is interesting)

2.7 So
for *r=1* we have (Does not depend of !!)

2.8 Besides
we have, as well as for :

3.1 Relation
between _{p}*(N*^{2}r) and _{p}*(r)*

Let *N*, *p1*, we obtain
:

Exceptional relation on which are based all these algorithms!!

3.2 According
to 2.4,
we have :

(Used series is called the series of Eisenstein *E*_{2}(q))

3.3 According
to 3.1,
we still have :

3.4 And
so for *N=p* we have according to 3.1 :

Now according to 3.1 and 2.6 we have :

*m*_{p,p}()=p

3.5 So we
find for *r=* and according to 2.6 :

_{p}*()=*

(Well, would not it be a good point of departure
for *=0,318*... !!)

4.1 Construction of the algorithm

We put

_{n}*=*_{p}(r_{0 }p^{2n})

*r*_{0} p^{2n }-> + when *n -> +*

so _{n} ~ because _{p}*(r*_{0} p^{2n}) -> according to 2.8.

We also put

*m*_{n}=m_{p,p}(r_{0} p^{2n})

This writing not being what we can call the most
practical, so we are going to find a relation between _{n} and _{n-1} knowing that :

_{0}*=*_{p}(r_{0}) for *r*_{0}=, so _{0}*=*.

The first part of the general study is ended. To build the algorithms, the series
consists in choosing one singular p and then to introduce modular forms *a(q)*, *b(q)* and *c(q)*. The existence
of a relation between *a(q)*, *b(q)*, *c(q)* and *a(q*^{P }), *b
(q*^{P }), *c(q*^{P}) supplies
modular equation

*a *^{p}+b ^{p}=c^{ p}

(No, no, it is not Last Fermat's theorem !!)

By defining :

, we
have *s*^{p}+(s*) ^{p}=1 and relations between *s*, *s** and supply the algorithm...

Application to the order 2 (p=2) :

5.1 Definitions :

According to 3.2,
we have *A*_{2}(q)=2P(q^{2})-P(q)=_{3}^{4}(q)+_{2}^{4}(q) according to 1.1 and 3.2

Let us put besides :

5.2 *a(q)=*_{3}^{4}(q)+_{2}^{4}(q)

5.3 *b(q)=*_{4}^{4}(q)

5.4 *c(q)=2*_{2}^{2}(q)_{3}^{2}(q)

Associated modular equation is :

5.5 *a*^{2}=b^{2}+c^{2}

With 3.1 we have :

5.6

Then according to 5.2 à 5.5, we find (not easily) :

5.7 and

5.8 *a(q)=a(q*^{2})+3c(q^{2})

By posing *s* and *s** as defined higher, we deduct from it :

5.9 and

5.10 ,
good...

We have with 5.5 :

5.11 *s*^{2}+(s*)^{2}=1

According to 5.9 and 5.6,
we have :

5.12 m_{2,2}(r)=(1+3s(q^{2})) (not too difficult, that!)

And, with 5.10 and 5.9

On the other hand, according to 3.4 and 5.12, we have

5.13

Now, of 3.4 and 3.5,
we know that _{2}*()=* and *m*_{2,2}()=p=2

and according to 5.12, *2=1+3S(2)* so *S(2)=*.

5.14 One has then _{n}*=(4*^{n})

5.15 and *s*_{n}=S(4^{n})

So, finally, we have _{0}*=*, *s*_{1}=S(2)=,
and

and with 5.13,

_{n}*=(1+3s*_{n})_{n-1} - 2^{n-1}s_{n}

Where from we obtain (finally!) the algorithm :

That is the most frustrating in this proof, it
is the simplicity of the principle and the parallel horror of the calculations that
does not appear here because the most difficult results are admitted...

Concerning series, I have the general formula of forming these series and the principle,
but not having understood everything, I let you to go to watch the site of Borwein
:

First series corresponds to the case *t=427* and the
second in the case *t=1555*.