Science, Maths & Technology

# Patterns and Landscapes

Updated Thursday 15th September 2005

Marcus du Sautoy delves into the work of Gauss and Riemann, the two mathematicians who started to discover the order behind prime numbers

I always feel a mixture of relish and dread when my neighbour at a party asks: "So what do you do?" Finding out that I am a mathematician, most people think that I must sit in my office all day doing long division to lots of decimal places. But that misses the true essence of what it means to be a mathematician.

Above all else, for me, a mathematician is a pattern searcher. Maths is about finding patterns in the chaos of numbers that surrounds us – to find the music which binds all these numbers together. And of all the numbers in mathematics, it’s the primes which offer the greatest challenge of all to the pattern searcher.

As you count through these indivisible numbers it is extremely difficult to predict where you’ll find the next prime number. Like the stars in the night sky, the primes appear scattered randomly through the universe of numbers. Some are clustered close together – others are flung far apart. There just doesn’t seem to be any pattern. It doesn’t make sense. And if there’s one thing a mathematician craves, it’s pattern and sense.

This problem of the pattern - or lack of pattern - of the primes has been like a magnet to perplexed mathematicians ever since the Ancient Greeks first proved they go on for ever. But it has outwitted the greatest mathematical minds of the last two thousand years.

It wasn’t until the late 18th century that a breakthrough finally happened. It was made by a 15 year-old German boy, who grew up to be one of the greatest mathematicians who ever lived: Carl Friedrich Gauss.

It was a present he was given for his 15th birthday that was to change the course of mathematical history. The present was a book of mathematical tables. At the back of the book was a list which began to obsess the young Gauss: a table of prime numbers. He spent hours pouring over these tables, trying to force them to reveal their secrets. His efforts eventually lead to an amazing discovery.

Faced with this table of unpredictable primes, Gauss executed one of the classic moves in the mathematician’s arsenal: if things get too complicated, do some lateral thinking. Look at the problem in a new way. Ask a new question. Instead of attempting to predict precisely which numbers are prime, Gauss asked instead how many primes there are.

Gauss started counting how many primes there were up to 10, up to 100, up to 1000, and so on. As he counted higher and higher, the primes seemed to be getting rarer and rarer. But was there some way to predict how they thinned out? As he counted his primes he realised he could calculate the probability of getting a prime. For example, there are 25 primes up to 100. That means there is a 1 in 4 chance that a number between 1 and 100 was prime. But between 1 and 1000, there is only a 1 in 6 chance of a prime occurring. Perhaps Nature chose the primes using a set of prime number dice.

But could Gauss predict the number of sides on the dice as Nature chose bigger and bigger primes? As he counted higher and higher, working out the prime probability, Gauss began to see a pattern emerging. Despite the randomness of the primes, a stunning regularity seemed to be looming out of the mist.

As he added a nought, Gauss realised that the proportion of prime numbers was decreasing by the same amount every time – by about 2. So from 10,000, to 100,000, to 1,000,000, the chance of getting a prime decreases from 1 in 8, to 1 in 10 down to 1 in 12. It was as if the primes around 10,000 were chosen using an 8 sided dice whilst those around 1,000,000 were chosen using a 12 sided dice.

Using his guess at how the primes thinned out, Gauss was able to estimate roughly how many primes you would expect to encounter the further you counted through the universe of numbers. For example, in the numbers from 1,000,000 to 1,000,120 the 12 sided prime number die would predict 10 primes in this region. But it was not an exact formula. And if it’s one thing mathematicians crave it’s precision.

Mathematics in Germany during this period was undergoing a revolution. Maths had traditionally been seen as a tool for calculation, a servant of the other sciences. In Paris, maths built ships and cannons. But in Gauss’s Germany the subject was changing beyond recognition. Maths was becoming abstract – an almost philosophical pursuit rather than a practical one. Mathematicians were entering an alternate, imaginative universe, filled with strange new forms of geometry and numbers.

In 1859, whilst exploring this new world, Gauss’s student, Bernhard Riemann, made the breakthrough that has galvanized mathematicians ever since. He found a way to go one better than Gauss and get an exact formula for the number of primes. Riemann’s formula contained information about precisely how the primes are distributed. Riemann found his formula after stumbling through a strange mathematical mirror which took him from the world of numbers into this strange new land of abstract geometry.

On the other side of this mirror Riemann discovered a mysterious mathematical landscape whose contours held the secrets of how Nature chose the primes. In particular, the points at sea level in this landscape, the places where the landscape has height zero, gave him vital new information about the primes. Mathematicians call these points the zeros.

Riemann realised that the precise location of each zero gave him information that could be used to correct Gauss’s guess and so get an exact formula for how the primes are distributed. He found that each zero could be used to produce a musical note whose vibrations perturb Gauss’s guess, correcting a little more each time. And it’s the combination of all the notes produced by the zeros which combine to create a music which corrects Gauss’s guess completely, giving the exact number of primes. It was as if Gauss had heard the dominant theme of the music of the primes but Riemann had uncovered the complete score that explained how these prime notes were really put together. The pitch and amplitude of each note was determined by the location of the corresponding zero in Riemann’s landscape.

Here was the treasure map that would explain the enigmatic primes. If you wanted to understand the primes, it was these points at sea-level in Riemann’s landscape that you really had to navigate. Uncovering such an unexpected connection between two different parts of the mathematical world is one of the biggest buzzes in a mathematician’s life.

But then Riemann noticed something even more incredible. As he plotted the locations of the first 10 zeros he noticed a rather amazing pattern beginning to emerge. They weren’t scattered all over the place but seemed to be lining up in a straight line running through the landscape. Riemann couldn’t believe it was just a coincidence that the first 10 zeros were on this line. He proposed that all the zeros, infinitely many of them, would be sitting on this straight line. This conjecture has become known as the Riemann Hypothesis.

But what did this amazing pattern mean for the primes? If Riemann was right about the zeros being all on this straight line it would mean that Gauss’s guess was far more accurate than even Gauss had imagined. It would imply that the prime number dice were not biased in any way but were distributing the primes as fairly as possible throughout the universe of numbers.

For me Riemann’s discovery is saying something rather beautiful about the music of the primes. If all the zeros are on the line, as Riemann proposed, it would mean there is a subtle balance between all the musical notes emanating from the zeros. But If Riemann was wrong and there was a zero off the line, then it would be like listening to a group of musicians playing and then suddenly a tuba comes in and drowns out all the other instruments. The louder the tuba, the bigger the error made by Gauss’s guess.

But Riemann didn’t believe there was a zero off the line. He was convinced that however far you travelled through the landscape, all the zeros would be on the line, implying that Gauss’s guess was always very close to the true number of primes.

However, Riemann could not prove his conjecture about the location of the zeros. For 150 years mathematicians have tried to prove Riemann right.

So many mathematical questions depend on knowing how the primes, the atoms of arithmetic, are distributed throughout all the numbers. Riemann’s Hypothesis would answer that key point on distribution. Solving the Hypothesis would be the Rosetta Stone of mathematics. It holds the prospect of creating a whole new language to be able to talk about these numbers.

But it isn’t only mathematicians who are interested in cracking Riemann’s riddle. Prime numbers are now at the heart of internet security. Every time we send our credit cards across the web we are using prime numbers to keep the transaction secure. The system capitalises on the difficulty of working out how to crack a number into its prime building blocks.

Every number is built by multiplying prime numbers together, just like every molecule is built from atoms in the Periodic Table. The chemists have invented a wonderful machine called a spectrometer which can take a molecule and tell you the atoms it is built from. What mathematicians haven’t been able to do is create their own prime number spectrometer which will take a number and tell you quickly the primes that built it.

Such a machine, if it were ever built, could crack many of the codes being used on the internet. It’s possible that a proof of the Riemann Hypothesis might give us sufficient new insights into the primes that it would give us a blueprint for such a machine.

Given the importance of the primes to mathematicians and e-commerce, it’s perhaps not surprising then that a businessman has recently offered a million dollar prize for a solution.

But mathematicians aren’t interested in the money or cracking codes. Most would sell their souls for a proof of the Riemann Hypothesis. It has become the Holy Grail of mathematics and whoever does finally prove the Riemann Hypothesis will be remembered forever as the mathematician who made the primes sing.

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