Although he passed away almost a century ago, on April 26, 1920, Srinivasa Ramanujan, the maverick Indian mathematician, has been a source of inspiration for several generations of mathematicians after him. Two recent papers co-authored by Indian students, Vanshika Jain and Kapil Chandran on fractional partition functions explore the charming insights of Ramanujan.
Walking along the path paved by Ramanujan, these young researchers have “extended the theory of partition congruences to all the fractional partition functions,” says Ken Ono, a Japanese-American mathematician, currently vice president of American Mathematical Society, who is widely known for his association with the making of the 2015 biopic on Ramanujan, ‘The man who knew the infinity’.
“Many of Ramanujan’s conjectures remain open, and his famed notebooks still contain secrets and ideas waiting to be uncovered. Our work focuses on the seemingly simple, yet unexpectedly complex theory of partitions” says Kapil Chandran, researcher, department of mathematics, Princeton University, USA.
What is Partition functions?
Let’s take number two.
How many ways can you write this as a sum of smaller natural numbers? In mathematician’s language, how many ways you can ‘partition’ a natural number? In the simple case of two,
it means we can ‘partition’ 2 in two different ways
What about three?
In the definition of partitions, the order does not matter; 2+1 and 1+2 are the same partitions of 3. This is how we can partition 3 in three different ways.
Before you hasten to conclude 5 can be partitioned in five different ways, you must know in fact, five can be partitioned in seven distinct ways. If only the number theory was that simple!
Mathematicians write this into a notation p(5) = 7.
We should read this ‘notation’, as ‘partition function of 5 is equal to seven’. Hence partition function relates the two quantities, a non-negative whole number and the number of ways it can be partitioned.
and so on.
Can you guess partition function of 20? One may think that it may be something like 50 or 100 or say 200. But one can partition twenty in to ‘627’ distinct ways.
Any guesses for partition function of 90? It is ‘56634173’!
As the number increases, the number of ways it can be partitioned show a surge in a mindboggling way.
The number of partitions of natural number series can be written as a series. The partition function series is:
1, 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, 5604, ... and so on.
Though seemingly elementary, it is very difficult to calculate p(n) as n grows larger.
Discovering hidden beauties
Look at the following series.
0,1, 2, 4, 9, 16, 25, 36, 49, 64, 81, 100 ...and so on.
Can you guess a pattern behind this series?
Yes, indeed they are squares of numbers 0, 1, 2, 3, 4, 5, 6 and so on. 1 squared is one, two squared is 4 and three squared is 9 and so on.
2, 4, 8, 16, 32, 64 ...and so on
is another simple series.
The next term is obtained by multiplying the previous term by 2.
Such patterns are easy and we often encounter them in our school mathematics.
Let us write the number of partitions of natural number as a series.
1, 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, 5604, ... and so on
Can you find any pattern?
They appear arbitrary. No easy pattern is discernible. Sure our high school mathematics is not enough. The insight of Ramanujan is called far.
Look at this series, in particular the numbers which are in bold.
1, 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, 5604,
All these numbers, 5,30,135 and so on are multiples of 5. The numbers 4, 9, 14, 19, 24, 29 and so on also have a pattern. These numbers can be written as 5k+4. When the k is 0, it is 4, when the k is 1 it is 9, when the k is 2 it is 14 and so on.
In a feat comparable to squeezing oil from a stone, in short Ramanujan noticed that p(5k + 4) is always a multiple of 5 for any number k. Ramanujan went a step further, he also showed that p(7k + 5) is always a multiple of 7 for all k. He published these observations as a paper in 1919. This is not all.
Ramanujan died tragically young, leaving sheaves of papers with many such intricate mathematical theorems and conjectures. Ramanujan’s mentor and lifelong supporter GH Hardy, a Cambridge mathematician, found an unpublished manuscript of the former, which proved yet another congruences -- p(11k+6) is always a multiple of 11 among these papers.
All these three patterns
p(5k + 4)
is always a multiple of 5
p(7k + 5)
is always a multiple of 7
is always a multiple of 11
These collectively are called as Ramanujan congruences on partition functions. Though congruences do not reveal the exact value of p(n), they give number-theoretic information on the exact value.
In the footsteps of Ramanujan
Along with parallelogram, the rectangle, the trapezoid, and the rhombus, the square is a special case of a quadrilateral, the four sided geometrical shape. Mathematics grows and gets deeper insights by extending the geometrical theorems of squares generalised to quadrilateral. For example, the total of its interior angles of any quadrilateral = 360 degrees.
Likewise, Ramanujan’s work on the classical partition function is specific to whole numbers, and mathematicians are attempting to extend it to other cases. “The alpha= -1 case is Ramanujan’s work on the classical partition function, while the research by these young students is an attempt to extended the theory of partition congruences to all the fractional partition functions,” says Ken Ono.
Partitions have deep connections to combinatorics, number theory, and representation theory, and therefore are of great interest to mathematicians. Moreover, going beyond, the number theory, also has applications in different areas like quantum field theory. Unearthing the hidden patterns help us understand the ways of working of nature and beauty of the cosmos.
The enduring spirit of Ramanujan
Srinivasa Ramanujan was one of the most talented mathematicians of all time. In spite of having little formal training in higher mathematics, Ramanujan developed advanced mathematical theories of his own in isolation, often retrieving difficult results proven only very recently. Ramanujan wrote letters to the other great mathematicians of his day describing his bizarre and puzzling insightful work. Thoroughly impressed by these letters, GH Hardy, a leading number theorist at Cambridge, invited Ramanujan to work with him in the UK. Over the next several years, Ramanujan made profound contributions to diverse areas of mathematics, including analytic number theory, elliptic functions, and infinite series.
Tragically, Ramanujan died in 1920 at the very young age of 32, leaving behind a rich legacy and web of conjectures that has occupied mathematicians for the past century. Many of these conjectures have finally been proven true long after his death. Of particular note is a conjecture on the size of the Ramanujan tau-function resolved in 1973 by the eminent mathematician Pierre Deligne, who won a Fields Medal for this work.
“Ramanujan’s mathematics continues to inspire talented mathematics students worldwide. The recent two papers on fractional partition functions extend Ramanujan’s mathematics to a wider class of objects and opens doors for new applications. Moreover, the authors are among the best undergraduate students studying mathematics in the US. They are star students at Penn State, Philips Exeter, Princeton, and MIT,” says Ken Ono. “We are humbled and excited to be researching areas of mathematics studied by Ramanujan himself,” said Dr Chandran.
While E Bevilacqua, department of mathematics, Penn State University, University Park and Y Choi, Phillips Exeter Academy, Exeter collaborated with Kapil Chandran, to produce the first paper, Jonas Iskander, Atlanta and Victoria Talvola, department of mathematics, Princeton University, collaborated with Vanshika Jain, department of mathematics, Massachusetts Institute of Technology, Cambridge, in the second paper.
(Jyoti Singh is a social media manager with India Science Wire and TV Venkateswaran is a senior scientists with Vigyan Prasar)