The Oracle / Ariel Bleicher

Mathematician Ken Ono solved age-old puzzles using insights hidden in the unpublished writings of the Indian genius Srinivasa Ramanujan

One Saturday morning in 1984, when Ken Ono was a high school student, he opened his family's mailbox in Baltimore and found an envelope as thin as rice paper, covered in colorful stamps. The envelope was addressed to his father, a reserved Japanese mathematician. Ono Jr. handed the mail to Ono Sr., who looked up from the yellow pad on which he always scribbled equations and put down his ballpoint pen. Gently, he unzipped the sealed envelope and unfolded the letter inside.

"Dear Sir," he began. "It came to my attention... that you donated money to erect a statue in memory of my late husband. I'm happy about that." The letter was signed "S. Janaki Amal,” who the red ink inscription at the top of the letter identified as the widow of “(the late) Srinivasa Ramanujan (mathematical genius).”

This was the first time young Ono had heard of the legendary Ramanujan. Ramanujan, a self-taught mathematical genius from India, made enigmatic arguments about a century ago that "seem as if one could hardly believe their correctness," once wrote Godfrey Harold ("GH") Hardy, the renowned British mathematician who worked alongside him. His work inspired entirely new mathematical branches and hinted at theories that in several cases earned their inventors the Fields Medal, the mathematical equivalent of the Nobel Prize.

Ono, who eventually became a mathematician himself, and is now a professor of number theory at Emory University, saw no reason during his studies to pay special attention to Ramanujan. As far as he was aware, the "mathematical genius" left behind no insights concerning Ono's unique expertise in number theory, modular patterns: two-dimensional abstract objects that mathematicians hold in high esteem for their remarkable symmetry.

Ramanujan re-emerged, in a big way, into Ono's life in 1998, when he was 29. Mathematician Bruce K. Berndt of the University of Illinois at Urbana-Champaign was then preparing an anthology of Ilavi's work, and came across a manuscript that had received almost no attention. The article dealt with modular patterns, so Berndt sent Uno a digital scan of it, thinking he might be able to decipher some of the strange arguments that appeared in it.

After Ono had read two thirds of the text, he stopped. In neat letters like a schoolboy, Ramanujan wrote down six bold mathematical statements that seemed completely outlandish to Ono, even though they touched on his area of ​​expertise.

Ono was a thunderbolt. He was convinced that the statements were false. "I looked at them and said, 'No way. It's bullshit.' "

His initial inclination was therefore to try to prove Ramanujan wrong.

The distribution plan

How did Ramanujan arrive at most of the mathematics he wrote? This question is shrouded in mystery. He taught himself from an old-fashioned English textbook, and in his mid-20s, while working as a government official, began sending his ideas in letters to mathematicians in England. He received one and only answer. This response came from Hardy, who had already begun to gain a reputation as a professor, and invited Ramanujan to come work with him at Cambridge. After only three years at Nicher, Ramanujan fell ill during the food rationing period introduced in the First World War. Emaciated and feverish, he returned to India and died in 1920 at the age of 32.

Besides 37 published articles, Ramanujan left behind a small library of letters, unfinished manuscripts and three leather-bound notebooks. Hardy and others examined them and found that he had rediscovered classical theorems, laws describing how numbers behave, first put down in writing by mathematicians who were the most senior in their field. Not this one either, Ramanujan noticed additional patterns, which no one had seen before. Skilled mathematicians should know how to substantiate each finding with a proof, a series of logical arguments that will convince their colleagues of its truth. But Ramanujan did not trouble himself. He filled pages upon pages with long lists of theorems and calculations he performed in his head or in chalk on a blackboard, rarely pausing to explain how he arrived at them. The three notebooks alone contain more than 3,000 conclusions about the nature of numbers, which mathematicians have labored to prove or disprove since Ramanujan's death.

Berndt began digging in the Ramanujan archive in the 70s. The subject continued to preoccupy him more than twenty years later, when he came across the manuscript that contained the six statements that caught his attention, statements that Ono was determined to prove wrong. These statements drew lines between modular patterns and so-called divisibility numbers: a series of integers that represent all the ways in which smaller integers can be added to get the number you started with. Division numbers come from a division function, which like any function describes a relationship between two things: it takes a certain input, x, and retrieves the corresponding output, f (x). the distribution function, p (n), counts the combinations of positive integers that add up to a given integer, n. For example, the function p(4) Gives the number 5: 1+1+1+1, 1+1+2, 2+2, 1+3, and 4.

The division function and the numbers it produces may seem simple and straightforward, but theorists have struggled for centuries to find patterns among these numbers so they can predict them, calculate them, or relate them to other functions and theorems. Ramanujan made one of the first real breakthroughs. He and Hardy built together a method for finding a fast approximation of divisor numbers. To test the accuracy of their approximations, they convinced a retired British gunner who was also a calculation wizard, Percy Alexander McMahon (aka Major McMahon), to calculate the first 200 division numbers by hand. Ramanujan and Hardy's approximations turned out to be impressive in their accuracy. More importantly, perusal of McMahon's list led Ramanujan to one of his most famous diagnoses. McMahon arranged my values p (n), Starting from-n=0, in five columns. Ramanujan noticed that every value in the last column, that is, every division number starting fromp(4), is divisible by 5, and he proved that this pattern continues forever. It was a shocking discovery. Remember, divisions are aboutConnection Numbers. No one imagined that they had properties that involved division.

Ramanujan saw that there were several other patterns similar to this. He proved, for example, that every number divisible by seven, starting withp(5), is divisible by 7. Similarly, any number divisible by eleven starting fromp(6) is divisible by 11. And mysteriously, these "Ramanujan congruences" stop there. "There seem to be no other equally simple properties for any other modulus involving prime numbers, except these," Ramanujan wrote in a 1919 paper, referring to the prime numbers 5, 7, and 11.

After Ramanujan's death, mathematicians wondered whether divisions might have some not-so-simple properties, and tried to find them. However, by the end of the 90s they were unable to find more than a handful of additional congruences involving apparently random prime numbers or their powers, such as 29, 17³ And 23⁶. They began to suspect that such patterns might be unpredictable and very rare.

But after Uno confronted those six forgotten statements from Ramanujan's manuscript, it dawned on him to his astonishment that these suspicions might be completely wrong. Mathematicians have long believed that partition numbers are associated with only a small subset of modular patterns. To Ono's surprise, Ramanujan's six statements connected the two fields in a profound way that no one had anticipated before.

Ramanujan did not record proofs, so Ono was unable to directly identify errors in the reasoning of the elevation. And so he decided to insert some numbers into the formulas that Ramanujan included in his statements, hoping that these examples would reveal any flaws. But the formulas worked every time. "unbelievable!" Ono said to himself. He realized that Ramanujan was probably right "because it's impossible for someone to be creative enough to come up with something like this and have it be true 100 times, unless he knows why that formula is true, always." Then he closed his eyes and thought with all his might what none but Ramanujan could understand.

Ono knew that in modular patterns "lots of congruences are scattered", the same patterns of division that Ramanujan found some of among the numbers of division. As Ono delved into the six statements, it occurred to him that if he thought of the division function as a modular pattern in disguise, he could show that they were true.

And another thought came immediately after: he realized, laughing out loud, that with a few adjustments, the theories he himself had developed about modular patterns could be powerful tools that would not only confirm Ramanujan's genius, but also help lift the lid on deep secrets. Regarding the distribution function. "It was like getting a cool new telescope," Ono recalled. "As soon as it is in your hands, if you start scanning space - when in this space the stars are the numbers of division - you will see that there are many, many galaxies."

Thus, Ono was able to prove that distributive congruences are not rare at all. Mathematicians assumed that there were only a few congruences after 5, 7 and 11. But in fact, as Ono discovered, there are an infinite number of them.

Ono's colleagues hailed his discovery as groundbreaking. But he was not satisfied. Although he could prove that distributional congruences are found in every corner, he could not say where they can be found. If you put the division numbers in order, you might want to know when a congruence will arise. If you have seen one congruence, can you predict when you will see the next one? Ono had no idea.

When Ono encounters a challenging problem, he refuses to grind it over and over in his head until it becomes as hard as old chewing gum. He prefers to catalog it in some corner of his head, alongside other unsolved problems, until it resurfaces. The problem, how to predict distributive congruence, lay dormant for five years until his postdoctoral fellow, Zachary A. Kent, came to Emory in the spring of 2010. It just popped up one day during a conversation, and soon they were talking about it all the time, in their offices, while Drinking coffee and taking a long walk in the woods north of Atlanta.

Little by little, they built in their mind's eye a winding and tangled superstructure into which the division numbers could be arranged in an orderly fashion. They discovered this organization using a theoretical tool, which mathematicians call an operator. The particular operator they chose takes some prime number (eg, 13), selects powers of the number (13², 13³ and so on), and breaks them down into the division numbers. Amazingly, the numbers he emits obey a fractal structure: they repeat in almost identical patterns at different scales, like the branches of a snowflake. This result shows that division numbers are not just a random series of numbers with random symmetries scattered within them without any order. Instead, these numbers have a "beautiful internal structure," Ono says, that makes them predictable and much more fascinating to explore.

Ono, Kent and their colleague, Amanda Folsom of Yale University, needed several months to refine their new theory. But finally they succeeded in proving that distributive congruences behave in a computable way. They exist for every prime number and for every power of a prime number. But after the number 11 the patterns become much more complicated, which may be why Ramanujan was never able to develop them.

Ono and his colleagues presented their findings at a symposium organized specifically for this purpose at Emory in 2011. After the symposium, Ono's inbox was flooded with well wishes. "This is a dramatic and surprising discovery," says George A. Andrews, a division expert at Pennsylvania State University. "I don't think even Ramanujan could have dreamed of her."

Beautiful answers

Exploring Ramanujan's insights led Ono to further discoveries that may one day prove useful in fields outside of mathematics. By combining Ramanujan's foresight with modern mathematics, Ono and his colleagues built powerful computational tools. Beyond advancing our understanding of pure mathematics, these tools could lead to better methods for encrypting computer data and studying black holes.

Ono developed together with Jan Bruinier, from the Technical University of Darmstadt in Germany, a formula for calculating large division numbers quickly and accurately, the holy grail that Ramanujan was unable to lay his hands on. Ono calls this calculating machine "The Oracle." According to him, the Oracle is not only capable of grinding divisions, but it can also be used for the purpose of studying a certain type of elliptic curves, geometric objects that look a bit like the surface of a clay.

Cryptographers (encryption experts) use elliptic curves to create algorithms for encrypting computer data. The success of these methods lies in their ability to create mathematical puzzles that cannot be solved in a reasonable amount of time. For example, a common algorithm called RSA [one of whose inventors is Professor Adi Shamir from the Weizmann Institute of Science - editors] is based on the difficulty of factoring a product of two very large prime numbers back into its factors. Newer methods use points on an elliptic curve, the relationships between which are even more difficult to identify. If the oracle, or similar discoveries, can shed light on other, even more elusive connections, cryptographers may be able to use this knowledge to build stronger encryption systems.

Ono's work also lifted the lid on one of the greatest mysteries in Ramanujan's mathematical legacy. Three months before his death, writhing in pain and feverish, he scribbled a last letter to Hardy in England. "I am very sorry that I have not written you even one letter until now," he wrote. "I recently discovered some very interesting functions, which I call 'theta-like' functions ... they fit into mathematics with a beauty that does not fall short of the characteristic of ordinary theta functions."

Theta functions are, essentially, modular patterns. Ramanujan hypothesized that it is possible to plot new, theta-like functions that are not at all like modular patterns but behave similarly for certain inputs called singular points. Near these points, the output of the function is inflated to infinity. Take for example the function f(x)=1/x, which has a singularity at a point x=0. As the input x getting closer and closer to 0, yes the output, f(x), increases to infinity. Modular patterns have an infinite number of such singular points. Ramanujan intuitively felt that for each and every one of these functions, there is a theta-like function that not only shares the same singular points with it but also produces an output that aspires to infinity at almost the same rate.

It was only in 2002 that a Dutch mathematician named Sander Zoegers was able to formally define the theta-like functions, using ideas formulated decades after Ramanujan's death. But mathematicians have yet to explain Ramanujan's statement that these functions mimic modular patterns at their singular points.

The mechanism behind Ono and Bruinier's oracle finally solved this riddle as well. Along with Folsom and Robert Rhodes of Stanford University, Ono used Oracle to derive formulas for calculating the outputs of theta-like functions as they approach singular points. Indeed, they found Ramanujan's conjecture to be correct: these outputs are remarkably similar to the outputs near the corresponding singular points in modular patterns. In one case, for example, the mathematicians found that the difference between them was very close to 4, a surprising and almost negligible deviation in this universe of infinite numbers.

Physicists have recently begun using theta-like functions to study a property of black holes called entropy, a quantity that measures how close a system is to reaching a state of perfect energy balance. Some scientists believe that formulas close to Ono's might allow them to test such phenomena with greater precision.

Ono cautions against the tendency to overemphasize the possible applications of his work. Like many other theorists, he believes that practical purposes are not what make such discoveries great. "Don't expect Ken's theorems to give us an infinite amount of green energy or find a cure for cancer or anything like that," says Andrews. Often, mathematical discoveries take an important place in science and technology only after they have simply been around for a few decades. It is difficult, and perhaps even impossible, to predict what this place will be.

Ono can still recall the delightful thrill he felt when he first saw the Ramanujan congruences written down, his father's steady hand tracing the foreign symbols on his yellow pad. "Why only three?" He remembers asking. "No one knows," his father answered him.

When Ono tells his story, he is sitting in his family's dining room in Georgia. On the wall behind him hangs a framed picture of Ramanujan's bronze prototype, commissioned for his widow with the help of $25 donations from Ono's father and hundreds of other mathematicians and scientists around the world. "Never, not even in my wildest dreams, did I imagine that a day would come and I would be able to say 'You know what, Dad? These congruences are not the only congruences, far from it.'"

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on the notebook

Ariel Bleicher is a freelance reporter working in New York City.

in brief

Srinivasa Ramanujan, a self-taught Iloy, filled notebooks with number theorems. These theorems, which were often shrouded in mystery, turned out to be true and eventually even spawned whole new fields in mathematics.

Now, at Emory University, Ken Ono and his colleagues have made some surprising discoveries with previously unrecognized insights from Ramanujan's unpublished writings.

Not only have these discoveries helped solve some big mysteries about mathematical machines called functions, they could also advance more secure ways to encrypt computer data and new approaches to studying black holes.

More on the subject

ℓ- Addic Properties of the Partition Function. Amanda Folsom, Zachary A. Kent

and Ken Ono in Advances in Mathematics, Vol. 229, no. 3, pages 1586-1609;

February 15, 2012.

Ramanujan's Mock Theta Functions. Michael Griffin, Ken Ono and Larry Rolen

in Proceedings of the National Academy of Sciences USA, Vol. 110, no. 15, pages 5765-5768; April 9, 2013. www.ncbi.nlm.nih.gov/pmc/articles/PMC3625272

The article was published with the permission of Scientific American Israel