Riemann's Zeta Function and 42

I normally don't post things that I didn't write but last few days I have been working on some maths to understand the Riemann Hypothesis. I don't mean that I am trying to prove the nontrivial cases. I just want to understand the problem and this takes time as I have to first understand Riemann's Zeta Function, Euler Product and their relationship. Anyway, the following article made me think about this and most probably will make me busy a few days more. The first video is a sort of a summary with a nice music accompanying the mathematical jargon. The rest of the videos are the ones I am trying to understand. The second and the third ones are easy but the following ones require real concenration and time. Not all of them I can concentrate on but I try my best to make some sense of the big convoluted picture. 

The article is taken from http://seedmagazine.com/content/article/prime_numbers_get_hitched/ 

In 1972, the physicist Freeman Dyson wrote an article called “Missed Opportunities.” In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell’s equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough—only announced by Einstein some 40 years later.

It is striking that Dyson should have written about scientific ships passing in the night. Shortly after he published the piece, he was responsible for an abrupt collision between physics and mathematics that produced one of the most remarkable scientific ideas of the last half century: that quantum physics and prime numbers are inextricably linked.

This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics.

Prime numbers, such as 17 and 23, are those that can only be divided by themselves and one. They are the most important objects in mathematics because, as the ancient Greeks discovered, they are the building blocks of all numbers—any of which can be broken down into a product of primes. (For example, 105 = 3 x 5 x 7.) They are the hydrogen and oxygen of the world of mathematics, the atoms of arithmetic. They also represent one of the greatest challenges in mathematics.

As a mathematician, I’ve dedicated my life to trying to find patterns, structure and logic in the apparent chaos that surrounds me. Yet this science of patterns seems to be built from a set of numbers which have no logic to them at all. The primes look more like a set of lottery ticket numbers than a sequence generated by some simple formula or law.

For 2,000 years the problem of the pattern of the primes—or the lack thereof—has been like a magnet, drawing in perplexed mathematicians. Among them was Bernhard Riemann who, in 1859, the same year Darwin published his theory of evolution, put forward an equally-revolutionary thesis for the origin of the primes. Riemann was the mathematician in Göttingen responsible for creating the geometry that would become the foundation for Einstein’s great breakthrough. But it wasn’t only relativity that his theory would unlock.

Riemann discovered a geometric landscape, the contours of which held the secret to the way primes are distributed through the universe of numbers. He realized that he could use something called the zeta function to build a landscape where the peaks and troughs in a three-dimensional graph correspond to the outputs of the function. The zeta function provided a bridge between the primes and the world of geometry. As Riemann explored the significance of this new landscape, he realized that the places where the zeta function outputs zero (which correspond to the troughs, or places where the landscape dips to sea-level) hold crucial information about the nature of the primes. Mathematicians call these significant places the zeros.

Riemann’s discovery was as revolutionary as Einstein’s realization that E=mc². Instead of matter turning into energy, Riemann’s equation transformed the primes into points at sea-level in the zeta landscape. But then Riemann noticed that it did something even more incredible. As he marked the locations of the first 10 zeros, a rather amazing pattern began to emerge. The zeros weren’t scattered all over; they seemed to be running in a straight line through the landscape. Riemann couldn’t believe this was just a coincidence. He proposed that all the zeros, infinitely many of them, would be sitting on this critical line—a conjecture that has become known as the Riemann Hypothesis.

But what did this amazing pattern mean for the primes? If Riemann’s discovery was right, it would imply that nature had distributed the primes as fairly as possible. It would mean that the primes behave rather like the random molecules of gas in a room: Although you might not know quite where each molecule is, you can be sure that there won’t be a vacuum at one corner and a concentration of molecules at the other.

                                        

For mathematicians, Riemann’s prediction about the distribution of primes has been very powerful. If true, it would imply the viability of thousands of other theorems, including several of my own, which have had to assume the validity of Riemann’s Hypothesis to make further progress. But despite nearly 150 years of effort, no one has been able to confirm that all the zeros really do line up as he predicted.

It was a chance meeting between physicist Freeman Dyson and number theorist Hugh Montgomery in 1972, over tea at Princeton’s Institute for Advanced Study, that revealed a stunning new connection in the story of the primes—one that might finally provide a clue about how to navigate Riemann’s landscape. They discovered that if you compare a strip of zeros from Riemann’s critical line to the experimentally recorded energy levels in the nucleus of a large atom like erbium, the 68th atom in the periodic table of elements, the two are uncannily similar.

It seemed the patterns Montgomery was predicting for the way zeros were distributed on Riemann’s critical line were the same as those predicted by quantum physicists for energy levels in the nucleus of heavy atoms. The implications of a connection were immense: If one could understand the mathematics describing the structure of the atomic nucleus in quantum physics, maybe the same math could solve the Riemann Hypothesis.

Mathematicians were skeptical. Though mathematics has often served physicists—Einstein, for instance—they wondered whether physics could really answer hard-core problems in number theory. So in 1996, Peter Sarnak at Princeton threw down the gauntlet and challenged physicists to tell the mathematicians something they didn’t know about primes. Recently, Jon Keating and Nina Snaith, of Bristol, duely obliged.

There is an important sequence of numbers called “the moments of the Riemann zeta function.” Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker’s Guide to the Galaxy.

It would also prove to be significant in confirming the connection between primes and quantum physics. Using the connection, Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence. Prior to this breakthrough, the evidence for a connection between quantum physics and the primes was based solely on interesting statistical comparisons. But mathematicians are very suspicious of statistics. We like things to be exact. Keating and Snaith had used physics to make a very precise prediction that left no room for the power of statistics to see patterns where there are none.

Mathematicians are now convinced. That chance meeting in the common room in Princeton resulted in one of the most exciting recent advances in the theory of prime numbers. Many of the great problems in mathematics, like Fermat’s Last Theorem, have only been cracked once connections were made to other parts of the mathematical world. For 150 years many have been too frightened to tackle the Riemann Hypothesis. The prospect that we might finally have the tools to understand the primes has persuaded many more mathematicians and physicists to take up the challenge. The feeling is in the air that we might be one step closer to a solution. Dyson might be right that the opportunity was missed to discover relativity 40 years earlier, but who knows how long we might still have had to wait for the discovery of connections between primes and quantum physics had mathematicians not enjoyed a good chat over tea. —Marcus du Sautoy is professor of mathematics at the University of Oxford, and is the author of The Music of the Primes (HarperCollins).

Originally published March 26, 2006

Black Swans in the eyes of a White Swan

The more I study mathematics of finance, the more I feel that what I learn does not make the life easier at all. The whole study of financial mathematics is based on the assumption that what happened in the past will repeat in the same way in the future. If a stock had high volatility in the past, then the price of the option for this underlying stock will be high because it carries high risk. However, no matter how complex mathematics we use to analyze the world of finance and economics, our predictions do not go beyond predictions simply because we do not know the future. However, with the help of complex mathematical jargon and intellectual arrogance, we can easily be convinced that some people with special skills have power to estimate the future movements of the market and can help you make money from betting against the improbable ones.

Nassim Nicholas Taleb’s book “The Black Swan: The Impact of the Highly Improbable” is all about the criticism of over-mathematisation of the finance and economics. He starts his book with some autobiographical stories; a childhood in Lebanon, war time stories, advantage of having a Levantine heritage and ending up at prestigious Wharton Business School. Of course, he criticizes his formal schooling as it is largely based on Gaussian Bell Curve and Markowitz Portfolio Theory. His main attacking point in the entire book is Gaussian Distribution (Normal Distribution) which is not capable of catching the black swans (extreme rare events) therefore it is useless because history is made of black swans, not the white ones. Revolutions, uprisings, natural disasters, wars, economic recessions are all examples of these black swans which cannot be predicted through conventional statistical tools. Predicting the normal is nothing according to him as it can be done by everyone and it does not give any edge to anyone who can do it. Plus, normal events does not cause any change in the history. They are left to the oblivion shelves of the past.

In one part he writes “Those who spend too much time with their noses glued to maps will tend to mistake the map for the territory.” meaning that our models should not be mixed up with the reality itself. They are only models and should not give us over-confidence. In fact, the book basically reveals that the biggest problem in today’s financial world is the overconfidence on the mathematical models and graphs as if they are really saying something about future. The scientific expressions and mathematical jargon make outsiders intimidated and ultimately keep them away from understanding the black swans.

I cannot say that Normal Distribution should be accused of the market failures or the mathematicians who help building all those financial formulas can be blamed. The problem is the profit-hungry brokers and CEOs who do not understand the assumptions behind all those theorems. In the world of mathematics, we need to platonify everything so that simplified versions can give us simple results. For example, infamous Black&Scholes option pricing formula is nothing but a one-page derivation of option price under the assumption that arbitrage-free and complete markets, efficient markets, constant volatility and interest rates etc. There are at least 10 assumptions for B&S to work efficiently and we all know that this is impossible to happen. Then the decisions which are heavily based on these mathematical formulae will eventually fail. But should we than blame the mathematicians for this? Or the men who take all those assumptions granted without considering that randomness is far from our comprehension and it cannot be tamed with a simple formula.

Besides this, normal distribution is a perfect tool to understand the normal events in our lives. Taleb talks about standard deviation with dismay but never mentions other risk measures which are much more useful to evaluate the risk of loss in the world of finance. For example, expected shortfall with a reasonable confidence level can easily catch the possible black swans in the normal set of events. Naturally, no one can predict the earthquake or the consequences of an earthquake. We can perhaps have large disaster reserves in insurance companies so that in the event of a big disaster the losses will be compensated without making people suffer for long time.

After reading the book, I can only conclude that today’s financial system which allows making huge profits through transferring risk is only a sham but nothing else. I can understand transferring the risk in the time of a loss as it does not benefit anyone but just covers the loss. This is why I think insurance companies should be strictly controlled by the government and should not be allowed to invest their reserves into risky portfolios in search of profit. At the end of the day, the reason for the existence of the insurance company is simply to help people who lose their property/loved one/vehicle/business etc so that society will keep moving without getting interrupted. Therefore, they are not supposed to make huge profits from the money they collect from people who are simply apprehensive about their futures.

I also found his comments on capitalism versus socialism very naïve and ill-informed. His defense for capitalism was nothing but a childish aphorism. Here it is:

“Capitalism is, among other things, the revitalization of the world thanks to the opportunity to be lucky. Luck is the grand equalizer, because almost everyone can benefit from it. The socialist governments protected their monsters and by doing so, killed potential newcomers in the womb. “

Well, here what he says is let the world live in its own luck, let the lucky ones make billions and the rest can live with the income under a dollar a day. He forgets that being unlucky brings suffering, brings tears and blood, brings misery to those unlucky people’s lives. So shouldn’t a responsible government think a solution for these unlucky people and elevate their life standards? If you live the entire world to the luck, what happens is 3% of world population will own more than 80% of the wealth and 97% will share the remaining 20%. Despite some social democracies in some countries, we still have an ongoing injustice and turmoil is most parts of the world. Imagine, when we leave it to mere luck (meaning pure greed) and watch it.

Here is another naive aphorism from the writer: “If people were rewarded strictly according to their abilities, things still be unfair –people don’t choose their abilities. Randomness has the beneficial effect of reshuffling society’s cards, knocking down the big guy.”

But he forgets that it is not the randomness, it is the anger of people which knocks down the big guy. We should neither support the big guy nor knocking him down (it costs lives of innocent people too) as we should only be against the existence of the big guy because by the time he is being knocked down (Think about Egypt: how many years did it take to dethrone Mobarek? How many people, journalists, intellectuals, writers have suffered under his regime?) so many would be perished in one way or other. What we need is not ups and downs, we need happy people with happy futures. We don’t need innovations of iphones or ipads to be happy. We need a secured future in which children can grow up without the fear of nuclear disasters or the toxics of e-waste. The world keeps dumping the e-waste to Africa now (I just watched a documentary about these children who collects copper and steel to burn in the garbage and sell for a few pennies, unlucky ones! huh?).

I would like to imagine if Mr Taleb was not a son of privileged Lebanese family, was not able to attend privileged schools to learn many languages and the tricks of stock markets, was not lucky enough to write books on unlucky people, then would he be thinking in the same way? Yes, he is lucky in his own terms but does this mean that he needs to be blind towards those who are not as fortunate as him. People who are born to poor families, born to an African tribe or to a family in a Libyan desert cannot see the world as he sees and definitely cannot agree with him. How unlucky Mr Taleb is that he cannot be with majority of the people but only supports a few privileged lucky ones. Changing his definition of the black and white swans, I can call him a lucky white swan who thinks that the unlucky black swans need to exist forever and interrupting nature's randomness will make things worse. Therefore, one should ask "why do we have empathy, why do we need to help others, why do have the feelings like mercy, compassion, tolerance? If nature made us greedy -I don't believe it despite the avalanche of discovery channel documentaries on wild animals- then why do we need to have peaceful societies, international rules, sports events? At the end, the book is full of aphorisms of a white swan on why black swans should stay as black. I agree with the criticism of epistemological arrogance of finance minded people but since the book does not offer any solution to the problems of the world other than the writer's over confidence on mathematics and statistics, I consider it as a failure, yet it sold millions of copies. So what? Harry Potter sold even more!!!

PS: He seems he does not know anything about Central Limit Theorem as he never mentions it. The reason Gaussian Bell Curve is used largely in today's world is not the variables fit to normal distributions. The reason is when we take a large sample from any set of data, the sum of the values in any possible sample can be approximated by a normal distribution. Therefore, nature does not need to be normal for us to communicate through normal distribution. I think, he is not a mathematician and he just learnt the formulae of financial mathematic to apply to his portfolio, I can forgive him. However, talking about great Mathematicians like Gauss, Galileo, Russell in a way he is better than them makes me angry. I am sure it will make any decent mathematician angry to hear those comments from a man who is only a stock broker and a rich man, yet calling himself philosopher.