Ramanujan's first paper on highly composite numbers

I am still reading The Indian Clerk. In between reads I look up relevant information from that period. Like when Ramanujan publishes his first paper in the book I tried to find what I could find of it through the Open University Library Services.

Early 1914 Ramanujan arrived in England. During the mornings he worked with Hardy and in the afternoons he followed lectures. One of the topics he was working on when he came to England were what Ramanujan called "highly composite numbers".

A highly composite number is a positive integer with more divisors than any positive integer smaller than itself.

The earliest paper of Ramanujan I could find was communicated by Hardy and published in the Proceedings of the London Mathematical Society; January 1915, Vol. 14 Issue: 1 p347-347, 1p.

If you have access this is a link: Highly Composite Numbers. ( pdf - 63 pages ).

page 2 of the paper

Update: 31/3-'11
The paper on highly composite numbers was not the first paper of Ramanujan. He published before he came to England in the Indian Journal of Mathematics. According to the "collected papers of Ramanujan", it was his 15th publication.

The Indian Clerk - ( Ramanujan Revisited )

I am reading The Indian Clerk by David Leavitt. Faction about the Hardy / Ramanujan collaboration. Reading a novel, through the appropriate reading protocol, gives an experience close to reality. Through reading The Indian Clerk you travel back in time to 1914 and meet Hardy, Littlewood and Ramanujan and a host of other very interesting people.

Hardy (l) and Ramanujan (r).

While in school Ramanujan spent more time on his own mathematics than on what he had to learn for his mathematics exams. As a result he was not drilled in doing proofs but doing mathematical research became natural to him.

Many schools include a research project as part of the graduation requirements for their mathematics majors. But most students are at a loss to create their own research questions, leaving this task to their advisors. It would be better if the student came up with their own research question that involved significant mathematical investigation and the creation of original mathematics. This is a daunting task: most graduate students are unable to do this, and rely on their advisors to frame a suitable area for investigation. The task is further complicated by the fact that many questions relating to undergraduate mathematics have “already been solved,” while many of the unsolved questions require so much specialized background to understand or so much existing research to review that the preparation needed to tackle the problem is itself a major project. - Jeff Suzuki in "But How Do I Do Mathematical Research?” on MAA website.

... the intensity of his interest in mathematics led him to pay scant attention to the other subjects in which he was obliged to show some facility. ... On each occasion, his interest in his own mathematical researches was so all consuming that he neglected his more quotidian studies, with the result that he failed his examinations and lost his scholarships. p118 ...no wonder he doesn't understand how to do a proof! p125 - The Indian Clerk

Ramanujan got -kicked out- of all schools he attended. Not fit for mathematics.

In the early 1900s, long before internet, good math books were rare in India or too expensive for Ramanujan. He learned his math from an encyclopedia type of collection of formulas and theorems. He must have re-invented the math that was not in that book because when he came to England he knew more or as much as Hardy, the leading Number Theorist of that time, but -in a different way-.

Long before Ramanujan arrived in England he was called the Indian genius. What is a 'genius' anyway? Newton was a 'genius'... Why didn't Newton share his invention of Calculus for such a long time? I suppose because ( knowledge ) power corrupts. Thank God Leibniz ( and others... ) independently invented Calculus around the same time. Science is for humanity not for the individual ( company ).

Ramanujan told ( to Eric and Alice Neville while they were interviewing him in Madras ) that his inventions were sent to him by the Goddess Namagiri in his sleep. Goddess?!, Red Alert! Stop.

There are two options here. Ramanujan's story about the Goddess Namagiri is true, or Ramanujan must have found a powerful method to do mathematical research of which he had no intention of sharing, like Newton, who used his then secret, Calculus to come with all sorts of amazing results. To declare someone a genius is giving up the search for his methods. How did Ramanujan do it? It's not in Ramanujan's notebooks let alone in Leavitt's book. Yet I tend to believe that if we want to come any closer to the Source of mathematics it is through Ramanujan.

Complex analysis videos

I am dividing my (study-)time between M381, M373, David Leavitt's novel The Indian Clerk (TIC) and self-study activities related to analytical number theory and complex analysis. ( And when I finished reading The Indian Clerk I'll start a book about the life of Kurt Goedel, or I might reread the book on Alan Turing. ) Reading, always keep reading, is my motto.

Excellent mathematicians work harder, have more luck in chosing their subjects and belong to more influential networks than less excellent mathematicians. Is that true? If we can prove so many theorems couldn't we have created one or two as well? Of course. Unless it was a 'theorem' of Ramanujan. His work, providing one has access to it ( read: understands it ) is of the jaw dropping class.

How did Srinivasa Ramanujan perceive the mest world?

In my self-study project I am struggling with the proof of the Prime Number Theorem as well as with understanding the Riemann hypothesis (RH). - In TIC it is G.H. Hardy's wish to prove the RH. At that point in his life he lectures at Cambridge. People in his surroundings we meet in the book are ( amongst others ) John Littlewood, Betrand Russell, Ludwig Wittgenstein and John Maynard Keynes. One day Hardy receives a letter full of mathematical scribblings. The impact of the magnitude of these scribblings reaches him, but slowly. The letter came from no one less than Srinivasa Ramanujan, an until then unknown Indian mathematician. Hardy discusses the letter with his young collaborator Littlewood and the story unfolds... An excellent companion to this reading adventure is Number Theory in the spirit of Ramanujan.



The main goal I have set for myself this -mathematical- year is understanding the PNT. The best way to reach that point was in my opinion studying Apostol's Analytic Number Theory. I make progress, but slow, although I am not terribly behind on schedule. The single most important effect though is that I now feel naturally motivated and ready to attack complex analysis. Last but not least: another set of video lectures on Complex Analysis by Bernd Schröder from Louisiana Tech University.

Lecture 1: Introduction

A measure I took based on my experiences with M208 last year is implementing a TMA-(latex-)code-freeze-date. I have set that date for next Tuesday when I'll start the check-double-checks. As M381 is a level 3 course I am content with the 65-sure I am at right now. Although there is no such thing as a 'sure' before the result is 'in'.

My URM emulator now automatically concatenates two URM programs. I want to automate substitution as well as primitive recursion. Although I am still struggling a bit with the manual implementation of a primitive recursive function at the deeper URM level. Knowing that this all leads ( and I am sure it does ) to understanding Goedels incompleteness theorems makes this all a worthwile adventure.

Terence Tao explains the music of the primes.

A talk on primes by Terence Tao. ( Tao won all the prizes there are to give. Charlie Eppes for real? A living Ramanujan? Decide for yourself. )

We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. - Tao

Time to start debunking Euler ?

Some names are only to be spoken of with respect.

Apple has been around a lot longer than Google. Apple created the i-Tunes software and related store where you can buy songs for only $0.99 and they created the i-Phone gadget. If you are a fan of that gadget I am sure you bought at least four of them in the last three years although the first one is still working it doesn't look that good compared to the i-Phone 4. Besides that Apple created billions of cash for Steve Jobs who has no intention to share it with the poor like Bill Gates from Microsoft does. - Google on the other hand created free software like gmail, google docs and google maps, a free feature we take for granted. Oh, and they created Google Search of course. Since a while, also thanks to Google, we have digital access to the books of Leonhard Euler ( and all the other great mathematicians of course ).

Have a look at the page where Euler defines the number Pi, here at Google Books. ( If you don't read Latin, why not use Google Translate? ) I checked the first 120 decimals of Euler's Pi with those from Mathematica 8 (2010) . They are exactly the same.

When I looked at Pi in 120 decimals I knew immediately that Euler did not make these calculations by himself. He just could not. OK, he was a genius but he did not have more -time- than a mere humanoid like us. How many hours ( if not days, i.e. check double-check ) would it have cost to do that calculation? It's not just -that- calculation. It's about the sheer size and depth of his legacy. Although Euler was still an active mathematician when he died at age 76, it is beyond the capabilities of one human being to create that in one lifetime.

Euler must have used students and computing staff to write his books. We know that he was blind during the last years of his life but even in that period he continued to publish. Perhaps 'Euler' was merely a brand. I don't know. Who was 'Euler'?

Perhaps the comparison of Euler with Ramanujan would be somewhat like a comparison of Rembrandt with van Gogh. We know for sure that everything we read in Ramanujan's notebooks is written by Ramanujan himself. I am not sure that everything in the books of Euler is written by Euler himself, the same is true for Rembrandt and are the cause of the fact that so many Rembrandt's turned out to be work of his students.

About students and tutors.

TMA Machine is another OU ( Mathematics ) student with a blog on http://catbear.wordpress.com/. In his right sidebar he maintains a list of OU Students with a blog. I thought about copying them to my bar but since the whole idea of the WorldWideWEB is linking ( and thus not copying ) I simply provide the link.

While reading some other blogs I realized that it is quite a handicap that Englush is not my native language. And worse I don't see much improvement either while reading back some of the posts I have written over the years. - ( Making a thought jump here. ) - Which proves my point that self-studying mathematics is nice but having a tutor looking at your work once in a while is the only -guarantee- that you will make any progress. If some English teacher would comment on the English in my blog entries I would have made some progress.

Ramanujan, considered to be one of the greatest mathematicians who ever lived,  was mainly a self-taught mathematician when he came to England to work with Hardy. It showed immediately: Ramanujan wasn't able to construct a simple formal mathematical proof. He could only progress in that area with the help of Hardy.

( testing ... LaTeX )

The probability of getting \(k\) heads when flipping \(n\) coins is:

\[\sum_{k=0}^{n} {n \choose k} = 2^n \]
\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]

An Identity of Ramanujan

\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} =

1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}

{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]

( ... more later !!! )

Mathematics in Movies ( 2 )

June, 2008 I wrote about mathematics in movies. As a comment on that post I received a link to this page with movies that are somehow linked to mathematics.
I got two math movies today. Fermat's Room and the Oxford Numbers. Both from 2008, maybe that explains why neither are on the page I mentioned. Haven't seen them yet though. Something for Christmas, perhaps.
I am not sure, but I think a movie about Ramanujan will be released in 2010. A must see. Ramanujan and Hardy in Cambridge. - Haven't watched Numb3rs in a while. I saved two seasons, for 'someday'. Well, maybe that day should be soon. Although Charlie is completely surreal, I mean he is an expert in -every- subject and has time to lecture, write books and be a full-time consultant to the FBI as well. Secret: I adopted one of his habits: I often use a noise-reduction headphone just to create silence around me, not to listen to music. It's great, you should try it while studying.

Ramanujan's magic square formula

Today I started to read the Ramanujan biography ( The e-book version, of course. ) The book looks promising. What was it like to communicate with someone gifted with such powerful mathematical insights? I am hoping ( a bit less after every failed try ) for a book that pulls me in that world of it's own. I have been there before on many occassions, while reading other books of course. Scientologists have a special word for it: exteriorizing.

The Man Who Knew Infinity.

At a very young age he designed the following formula for a 3 by 3 magic square:

C+Q | A+P | B+R
A+R | B+Q | C+P
B+P | C+R | A+Q

where A,B,C are integers in arithmetic progression and so are P,Q,R.

Try it, it works!

Mathematics in the Movies

Not counting Numb3rs there aren't many movies truly about math. As far as I know anyway. I know of the following movies ( in alphabetical order ):
- A Beautiful Mind
- Flatland
- PI
- Proof.
All these movies are fairly recent. The oldest is PI, from 1998. So that's encouraging. There is a movie about Ramanujan / Hardy in the works. Looking forward to that one.