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Researchers have finally solved the cryptic mathematical puzzle of Srinivasa Ramanujan. In 1920, Ramanujan wrote a letter to G.H. Hardy outlining several new mathematical functions that had never been heard of before, together with a theory about how they worked. These mathematical functions baffled mathematicians for more than 90 years. But new findings reportedly show that Ramanujan was right.
In the year 2012, during a conference at the University of Florida one of the Plenary Speakers, Ken Ono of Emory University, United States, told the audience that they had finally solved the problems from Ramanujan’s last mysterious letter. Prof Ono referred to the problem mentioned in Ramanujan’s last letter to Prof Hardy which had been open a challenge to the mathematicians for 90 years. Ramanujan's legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died. ’We proved that Ramanujan was right,’ Prof Ono said in his speech, ‘no one was talking about the black holes back in the 1920s when Ramanujan first came up with the mock modular forms, and yet, his work may unlock secrets about them.'
Mock theta functions
Ramanujan listed 17 examples of functions that he called mock theta functions. He also listed several other examples of the function in his notebook. Ramanujan used the term "theta function" for what today would be called a modular form. Ramanujan conjectured that his mock modular forms corresponded to the ordinary modular forms earlier identified by Carl Jacobi, and that both would wind up with similar outputs for roots of 1. Nobody at the time understood what Ramanujan was talking about.
Ramanujan's 'simple' pattern
It seems to be an easy problem to find out all possible that a number can be created by adding together other numbers. However, the solution to this leads to a greater understanding of 'partition numbers', a cryptic phrase Ramanujan used to describe sequences.
A partition of a number is any combination of integers that adds up to that number. For example, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1, so the partition number of 4 is 5; i.e. p(4) = 5. It sounds simple. The partition number of 10 is 42, while 100 has more than 190 million partitions! So a function for calculating partition numbers was needed.
Ramanujan was the first mathematician to seriously investigate the properties of this function. He sought a formula for p(n), one which describes the phenomenal rate of growth suggested by the table below:
n p(n)
1 1
2 2
3 3
4 5
5 7
.
.
50 204226
.
.
200 3972999029388
.
.
1000 24061467864032622473692149727991
If we arrange p(n) of first 30 numbers (i.e. 0 to 29) in five column form, we get the following table:
1 2 3 5
7 11 15 22 30
42 56 77 101 135
176 231 297 385 490
627 792 1002 1255 1575
1958 2436 3010 3718 4565
The striking feature of this table is that every entry in the last column is a multiple of 5.
Ramanujan's approximate formula, developed in 1918, helped him spot that numbers ending in 4 or 9 have a partition number divisible by 5. He proved for every non-negative integer n, that p(5n + 4) ≡ 0 (mod 5).
Ramanujan found similar rules for partition numbers divisible by 7 and 11. Without offering a proof, he wrote that these numbers had ’simple properties’ possessed by no other numbers. Later, similar rules were found for the divisibility of other partition numbers. Therefore no one knew whether Ramanujan's words had a deeper significance.
Ramanujan’s work on p(n) inspired research of modular forms. Theory of partitions has historically served as a “testing ground” for some of the deepest developments in the theory of modular forms.
Ramanujan’s last letter to Hardy
On 12 January, 1920, just three months before his death, Ramanujan wrote his last letter to Hardy. Ramanujan said in this letter:
“I am extremely sorry for not writing you a single letter up to now. I discovered very interesting functions recently which I call ’Mock’ ϑ-functions. Unlike the ’False’ ϑ-functions (studied partially by Prof. Rogers in his interesting paper) - they enter into mathematics as beautifully as the ordinary theta functions. I am sending you with this letter some examples.”
This letter contained 17 examples. Most of the surviving text of the letter, which included roughly 4 typewritten pages, consisted of explicit formulas for these 17 strange formal power series.
Ramanujan even divided these examples into groups based on their ‘order’, a term he never defined. Despite these formidable challenges, a few mathematicians such as G. E. Andrews, L. Dragonette, A. Selberg, and Watson investigated Ramanujan’s mock theta functions .
Despite the absence of a theory, or much less, just a simple useful definition of a mock theta function, these few early works bolstered the belief that Ramanujan had discovered something important.
Watson, in his own words: “Ramanujan’s discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. As much as any of his earlier works, the mock theta functions are an achievement sufficient to cause his name to be held in lasting remembrance. To his students such discoveries will be a source of delight and wonder until the time shall come when we too shall make our journey to that Garden of Proserpine (a.k.a. Persephone)”.
These 17 examples, together with five examples of mock theta from Ramanujan’s notebook have been related to a number of subjects including Lie theory, Modular forms, and Polymer chemistry. Despite this flurry of activity, the essence of Ramanujan’s theory remained a mystery. The puzzle of his last letter to Hardy, thanks to the ‘lost notebook’, had morphed into the enigmatic web of Ramanujan’s 22 mock theta functions. This strongly suggested the existence of a theory, and it also demanded a solution.
A new beginning
As Ken Ono said, “… Although Ramanujan’s last works provided the first examples of such forms, his untimely death and the enigmatic nature of his writings resulted in a great mystery. We will never know how he came up with the mock theta functions. We certainly cannot pretend to know what he fully intended to do with them. However, it is clear that he understood that the mock theta functions would go on to play important roles in number theory.”
Ramanujan's mock theta functions or mock modular forms is not only an important tool in analytic number theory, it is applied in several other areas of mathematics like, topological invariants analysis, Lie superalgebras —to name a few.
Ramanujan died before he could prove his theorem. But more than 90 years later, Ono and his team have proved that these functions indeed mimicked modular forms, but don't share their defining characteristics, such as super-symmetry. The expansion of mock modular forms helps physicists computing the entropy or level of disorder of black holes.
Rintu Nath
Rnath@vigyanprasar.gov.in