Warning !Genius !!!
Srinivasa Ramanujan
(1887 - 1920)
A few formulae (out of the many
possible ones)
By denoting (x)_{n} the value : (it's Pochhammer's symbol), we get :
Phew!
Slices of his life
With Ramanujan, we reach the quintessence of the study of Pi. He is the master of all
research in the 20th
century in this domain, let's not be afraid to acclaim it!
I could spend days talking about this unrecognise genius who
nevertheless left a magistal work still yet not understood in a few
domains. His life is furthermore a true novel....
But let us start by the beginning :
Srinivasa Ramanujan was
born the *22 decembre 1887* in the town of Erode in the south of
India in a poor family. His dad was accountant. His mathematic
precocity was quickly noticed and at seven, he recieved a bursary for
the secondary school of Kumbakonam (!). It is said that he would recite
mathematical formulae to his classmates and that he knew a surprising
amount of the decimals of Pi!
Since the start of his study on trigonometry, he discovered cos and
sin, found all the relation linking the two and showed to be very
dissapointed when he learned that they were already known!!!
When he was *12*, Ramanujan master a huge and dense book: *Plane
Trigonometry* by Loney.
When he was *15*, he got hold of *Synopsis of elementary
results in pure and applied mathematics* by G.S.Carr, a list of *6165*
theorems states often without proofs. We assume that it's from this
book inspired him abd his bad habit of not giving out proofs with his
results!
In fact at that period he was so obssesed with his research that he
failed all his exams!
Luckly, after his wedding in *1909*, he received a monthly sum
from a rich patron of the art passionate with mathematics (R. Rao) on
the recomodation of the indians mathematicians who appriciate his
discovery already written in what we commonly call his notebooks.
Gaining a stable job in *1912* as a beaurocrat at Madras'
counter, he was encouraged by his managers to send his result to 3 distinguished british
mathematicians, of which only G. Hardy replied to his letter dated *16/01/1913*.
In fact, when Hardy and his collegue Littlewood examined a few of the *120*
formulae and theorems send by Ramanujan, their conviction was made a
few hours later: they were looking at a genuis! (Hardy had build a
"scale of pure capacities" on which he situated himself at *25*,
gave *30* to Littlewood, and *80* to Hilbert, shinning
figure of the german mathematics from the start of the century.
Ramanujan was immediatly estimated at *100* !!!).
Hardy described later that the intellectual discovery of Ramanujan and
its consequences as the only "romantic" evenment of his life....
When he looked at Ramanujan's formulae, he was disconcert and had no
idea on how to proof the. But, he maintained, "they must be ture
because he they weree not, nobody in the world would have enough
imagination to invent them!"m
he made Ramanujan come to England and work with him for the next five
fruitful following years on the properties of several arithmetics
functions. Srinivasa became the first Indian to be member of the Royal
Society in 1918 and of
Trinity College (Cambridge).
Unfortunatly, and its such a shame, Ramanujan was strictly vegetarian
(due to a promise made to his mother!), and in England in the full blow
of war, his needs were hard to fufill... After the war, in 1919, he came back in India
seriously ill from a tuberculose and a lack of vitamines (We all know
how wet Brittain is!).
His work would stay of great quality despite his suffering, but he drew
his last breath the *26 april 1920*, he was only *32*.
A little anecdote very well known by
Hardy :
I remember once going to see him when he was ill at Putney. I had
ridden in a taxi cab number 1729
and remarked that the number seemed to me rather a dull one, and that I
hoped it was not an unfavorable omen.
"No" he replied "it is a very interesting number; it is the smallest
number expressible as the sum of two cubes in two different ways." !!
I then asked him if he knew what was the next one. He thought about it
for a minute and told me that he could not see any close one... In fact
, the next one is several thousands later!! (4104)
Around
Ramanujan was passoniated with pi. Many of his results involve our
favourite constant...
Ramanujan wrote down his work in his notebooks as I stated above.
Unfortunatly many formulae are written in with non-standards notation
and without any proofs. For more than 80
years, several mathematicians (Bruce Berndt for the moment) tried to
decipher his coded notebooks for the happiness of Science!
Because Ramanujan worked on the modular equations. But what exactly are
they?
I will 'borrow' a very clear example from the book *Les
mathématiciens* to clarify the definition.
A modular equation is an equation that satisfy a modular function *(q)* where the variable q intervine with
various power, for example *(q*^{n}). n denotes here the
order of the modular equation.
Consider for example the modular equation of *7th* ordre (*n=7*)
:
We then look for the solution to this
equation. For our case, we have:
(don't ask me how it was found...)
Up do here, no hints of Pi...
Well, we call for singular values from the values of the modular
function *(q)* which satisfy some extra
properties. For example, if we define for p integer:
Straight away we see that the bigger p is,
the smaller the exponential and the product in *(q)* tends toward *16q*
So, if we take , we get a number
that coincide on the first decimals with *Pi* !
Of course, the number of decimals increase with *p*. We can see
the whole advantage in having a relation between *(q)* and *(q *^{p}), this last number
being closer to *Pi* than the first (because the exponential is
even smaller!)
The amazing thing about this theory, is that the singular value does
not depend at all on Pi even
with their definition
Ramanujan was a great specialist of those values and calculated them in
a remarquable way. In his lettler to Hardy, he said:
which allows us to get 20
decimals of *Pi*.
*K*_{240} allowed his to get the first million
decimals of *Pi* !!
But he never did any research on the algorithm that we could build from
them. The Borwein brothers took up that
idea, here is how they proceded.
Principle of the proof of the Borweinein
If we look at the Borwein's
proof for the algorithm of second order, we can see cleary that we
consider the modular equation of second degree:
*a*^{2}+b^{2}=c^{2}
with *a(q)=*_{3}^{4}(q)+_{2}^{4}(q), b(q)=_{4}^{4}(q), c(q)=2_{2}^{2}(q)_{3}^{2}(q) and the
Thêta Functions :
This equation lead to let and to find the true modular equation of
second degree useful here :
whose solution is well known according to
the definition of *s(q)* (in fact we started here from the
solution to get to the modular equation.)
The initial value *s*_{1} correspond to a first singular
value, the sequence *s*_{n} hence allows to calculate a
sequence of singular values.
The rest now like above consist of finding a way to come back to Pi, since logarithm did the job in
the first example. The role here is played by the _{p}*(p*^{2}r), and
then we find from it an algorithm. the justification is given by the
apparition of the function in Legendre's
equation.
In all, a lovely little theory....
While we're talking about Ramanujan's
formulae :
This page is a little bit special as it does
not contain any proofs. I just reproduces the intermediate steps of the
detailled calculation on the Borwein's
canadians page. a part form the general
formula of the formation of series of type Ramanujan found by the Borwein and written on my page dedicated to
them, I prefer allowing you to see their site....
Other results by Ramanujan :
So, what else ?
There's first of all numberous approximation of *Pi* that he
found in a prodigious way! They are written in the page on approximations of Pi as well as the famous
result showing that is close to an
integer...
Otherwise, there doesn't unfotunatly exist much information on the
results of Ramanujan. Nevertheless, here is what I could find:
Prime numbers (function *Tau* of
Ramanujan) :
Be so that , then for GCD(n,n')=1,
This first property allows to calculate the function *Tau* for
all the product of prime numbers knowing the value *Tau(p)*.
And since the naturals can be expressed as the product of prime
numbers, the function *Tau* is completly evaluated on *N*
if we can evaluate it for the powers of prime numbers, which is what
the following theorem does by recurrence:
This function *Tau* has quite a few
more properties:
There exists some congruence relations for 7, 23 and 691. For instance, .
Moreover, Ramanujan conjectured that for p prime.
Note that the relation of congruence is only true if *k* is
without quadratic remainders modulo* 23*, that is that there
exists no integers *x* such that *x*^{2}=k[23].
Since there are *(p-1)/2* quadratic remainders for *p>2*,
we can deduce that on average, one natural integer *N* in two is
such that *tau(N)* is divisible by *23*.
Pierre Deligne (belge and not french) showed in *1971* that the
conjecture above was one of the consequence of the conjectures by Weil.
And since he obviously didn't want to leave it at that, he proved them
in 1973, which was
responsible in part for his Fields Medal in *1978*.
And we also have :
where is the sum of eleventh power of the dividers of n.
For more information on sigma, cf.A013959
Landau-Ramanujan's
constant :
The proof is self obvious if you do a variable change *t=x-1*,
then write the finite expansion of *Ln(1-t)*, and by justifying
the change between sum and integral. I did not look for *a*_{2},but
it seems completle doable!
Spread of prime numbers :
where c is the root of the function Li defined by and is the Möbius function defined by =1 if k=1, (-1)^r of k is the product of r distinct prime numbers, 0 otherwise.
For more information on the function pi(x),
cf.A000720
on Möbius' function, cf.A008683 and the relation between Möbius and Pi, cf arithmetic functions page.
Equations on the theory of numbers :
Ramanujan gave himself the problem to find the solutions to the
equation :
*2*^{N}-7=X^{2}
The computers have been able to find
up till *N=10*^{40} but only the solutions *(N=3,4,5,7,15)*
were confirmed. We have in fact proved recently that they are the only
valid !
For more information on this equation, cf.A060728
An other problem by Ramanujan ? But of course :
Find for example all the solutions of *n!+1=x*^{2}
(I unfortunatly don't have the list...)
Thanks to Christian Radoux
for his precision on the function *Tau*
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