Based on a talk about the work of Bernhard Riemann given at a
Sareoso Meeting in Birmingham in 2007
From time immemorial people have been fascinated by the prime numbers. These are the numbers such as 3, 5, 7, 11, 13 and 17, which are only divisible by 1 and themselves. If you go through the counting numbers clapping on the primes you will find that there is no regularity to your clap. Their distribution is apparently random. Generations of mathematicians have studied the prime numbers attempting to discover whether there is a hidden order that underlies them.
In the 19th century a German mathematician called Bernhard Riemann (1826–1866), building on the work of his predecessors and in particular his tutor, Carl Gauss, broke new ground in the study of the primes. Gauss had discovered that there was regularity in the rate of change in the proportion of primes to other numbers; Riemann uncovered a deeper and more mysterious order. His explorations led him into a strange world based on complex numbers. This journey was to revolutionize mathematics, influence the development of e-commerce, computer sciences and quantum mechanics, and lay down challenges that are absorbing mathematicians to this day.
I am not a mathematician. I was so bad at mathematics at school and so hated it that my teachers completely gave up on me, but I found the story of Riemann’s work compelling. I am going to try and tell the story. School mathematics seemed to be largely utilitarian; this story raised fundamental questions for me about reality. I have tried to draw out some of these questions at the end of the paper. Also at the end are the names of two books which I would recommend to anyone who would like to read more on the subject, and the address of a website which goes into the more serious mathematics.
In order to understand complex numbers it is necessary to set them in context. Numbers fall into five categories. There are (1) the natural numbers. These are the simple counting numbers familiar to us from childhood; (2) the integers, which are the natural numbers, their negatives and zero; (3) the rationals, which are the fractions and decimals. These are the numbers concerned with the ratios between different magnitudes; (4) the real, which include all the previous categories plus the irrationals; and (5) the complex, which include the real and the ‘imaginary’.
There are an infinite number of integers and rationals. Although we cannot grasp such a concept we can at least get some sense of it. We can sort of imagine whole numbers and their negatives going on and on getting bigger and bigger (and computers can now generate numbers with millions of digits) and myriad fractions crowding every corner of creation. When the irrationals come along the mind begins to boggle. These slip and slide and prove ungraspable. We are going to attempt to get a handle on them in the interests of stretching the mind. This helps to make the mind more flexible to prepare it for contemplation of the imaginaries.
When rational numbers are expressed in decimal form a repeating pattern emerges. This can take the form of a single unit recurring endlessly, for example the fraction 1/3 in decimal form is writing 0.3333… and ½ is written 0.50000… or a pattern of units repeating as in the decimal form of 1/7 which is 0.142857142857142857…The irrational numbers have no repeating unit or pattern. No matter how far one calculates them the units continue to unfold without reaching a point where one can say the number ends. There are an infinite number of irrationals, just as there integers and rationals, of which the best known are phi () which is the Golden Section and has a decimal form (taken to six digits) of 1.618034, pi () which is concerned with the relationship between the diameter and circumference of a circle and has the decimal form 3.141593 and e which is concerned with growth and decay and is the basis of natural logarithms.
A Greek mathematician called Hippasus used Pythagoras’s famous theorem – the square on a right triangle’s hypotenuse equals in area the sum of squares on the two sides – to prove the existence of irrational numbers. To follow the logic of this proof we need first to check the theorem. In the following diagram a and b are of equal length and each 1 unit long.
So we then have an equation that looks like this
a2 + b2 = c2 which is the same as12 + 12 = 22 which is the same as 1 + 1 = 22
Therefore c = √2
Hippasus starts off his proof with the assumption that √2 is a rational number, which means that it can be expressed as the ratio between 2 natural numbers. So he says:
1. √2 = a/b (where a and b represent any natural number.)
An absolute principle of algebra is that things are always expressed in their simplest form. If there is a common denominator in a fraction between the top number (the numerator) and the bottom number (the denominator) then the fraction is cancelled out. For example, in the fraction 9/12 both top and bottom are divisible by 3 so the fraction is cancelled out to ¾. In the equation √2 = a/b the a/b is assumed to have been cancelled out to its lowest common denominator.
2. Again in the interests of simplicity, Hippasus adjusts the equation from:
√2 = a/b to 2 = a2/b2.
If an adjustment is made to one side of an equation it must also be made to the other side. In this case we squared √2 to get 2, so we also have to square a and b.
3. Again in the interests of simplicity and because multiplying is easier than dividing, Hippasus adjusts the equation from:
2 = a2/b2 to 2b2 = a2 x b2/b2
The b2 on the top and bottom of the fraction then cancel each other out so:
2b2 = a2 x b2/ b2 becomes 2b2 = a2
4. Since a2 = 2b2 then a2 must be even, so a must be even. Remember that we started with a/b representing the ratio between 2 natural numbers. Twice any natural number is an even number, for example 2 x 6 = 12 and 2 x 9 = 18 so we know that 2b must be an even number. Only even numbers have even squares. If you square an odd number it results in an odd number, for example 3 x 3 = 9 and 5 x 5 = 25 This means that in the equation 2b2 = a2 the a2 must be even because the 2b2 is even. Are you following so far?!
5. So, a can be written as a = 2n
(where n is used to represent any natural number.)
6. If a = 2n then a2 = 4n2 (because both sides of an equation have to be adjusted equally.)
7. Refer back to number 3. above, and you will see that a2 = 2b2 If a2 = 4n2 then 4n2 = 2b2
8. There is a common denominator here on both sides of the equation so we can simplify it to 2n2 = b2
9. By the same reasoning as number 4 (above) we know that b2 must be an even number so b must be even, and that gives us a problem! Remember that we started the proof by canceling out all the common denominators in the fraction a/b. So a/b was a fraction in its lowest terms. Now we are saying that both a and b are even numbers. The fraction a/b is therefore not in its lowest terms because they are both divisible by 2. Hippasus has proved that there is an inherent contradiction in the initial assumption. The square root of √2 cannot exist as a ratio of 2 integers. Therefore it has to be an irrational number.
Armed with the knowledge that (1) irrational numbers have no repeating pattern in their decimal form and (2) that c in our diagram is an irrational number, let us now see what happens when we extend the line marked b and then lower c as if it were the boom of a crane until it touches this line.
The square root of 2 in decimal form is 1.41421356 …. So where exactly does it meet b? All we can say is that length c meets bsomewhere between 1.4 and 1.5, or more precisely between 1.41 and 1.42, and more precisely between 1.414 and 1.415, and more precisely still between 1.4142 and 1.4143, and we can go on getting more and more precise about this illusive meeting point without ever finally pinning it down. We also know, through Hippasus’s Proof, that c does not exist as a ratio of two integers. So how can it be described as a length at all?
A length can only exist in relation to some other measurement. At this point my mind begins to do philosophical cartwheels. How can this magnitude be said to exist when it cannot be perceived in relationship to another? After all, we can only identify something because it is comparable with something else. And how can irrationals be said to have an identity when they have no boundary? Between one thing and another there is some kind of cross over where the first thing ends and the next begins, but the irrationals have a tail composed of an infinite amount of numbers after the decimal point. If we cannot see where they end, how can we see where they start? The impression I am left with is of an amorphous mass of illusive mathematical beings slithering around in the cracks between their measurable and rational relations like some strange kind of lubricant.
Setting aside these conundrums for a moment, let us now return to the subject of complex numbers. Mathematicians came up with the notion of complex numbers to solve the question ‘what is the square root of minus 1?’ The square root of any number eg. n, is that number which, when multiplied by itself, gives n. For example, when n = 4 the square root is 2. However, a positive number multiplied by a positive number gives a positive number, and a negative number multiplied by a negative number gives a positive number. So the question ‘what is √-1?’ would seem to be unanswerable. Mathematicians resolved this conundrum by saying let us assume that this number exists even though we cannot conceive of it from the perspective of the conventional rules of calculation. They called √-1 an imaginary number and gave it the symbol i. When i is combined with a real number (ie. integers, rationals and irrationals) it is called a complex number.
A German mathematician called Carl Gauss (1777–1855), who was a colossus in the history of mathematics, extended the concept of complex numbers by creating a 2 dimensional diagram called the complex plane. In the complex plane there are two axes, one representing the real numbers and the other representing the imaginary numbers that intersect at zero. Gauss realized that on this map there was a point that represented the solution to all possible equations that combined real and imaginary numbers. He used the complex plane to plot calculations in a graphic form as though they were coordinates on a map. In the following diagram the coordinates 7 + 5i is derived from (5 + 3i) + (2 + 2i)
Complex numbers behave in a different way to real numbers. They behave differently when used in calculations and when depicted graphically. Firstly, look at what happens in the following sequence in (when n = any whole number):
i = i
i2 = -1
i3 = -i
i4 = 1
i5 = i
i6 = -1
i7 = -i
i8 = 1
i9 = i
in = ..
To work through this you need to remember the rules about multiplying positive and negative numbers. i4 = i2 x i2 = (-1) x (-1) = 1 (because a negative multiplied by a negative gives a positive) and i9 = (i3) x (i3) x (i3) = (-i) x (-i) x (-i) = (i2 ) x (-i) (positive multiplied by negative gives a negative) = (-1) x (-i) = i (negative multiplied by negative gives positive)
If you look at the answers in this series it is clear that a pattern emerges. i is followed by -1 and then –i and then 1, and the same sequence then goes on repeating. Such a regular pattern would not have occurred if a real number had been used in the place of i. For example, if the number had been 2. In passing, it is interesting to note that i is not only the square root of –1 but is at the root of every fourth number in the sequence.
A century before Riemann, the great Swiss mathematician Leonhard Euler (1707 – 83), had discovered that interesting things occurred when an exponential function was fed with complex numbers and the results plotted graphically. An exponential function is a function that increases at a rate proportional to its value at the present time. When fed with real numbers these calculations produce a curve that climbs steeply without deviation (ie. which climbs ‘exponentially’). Euler discovered that when exponential functions are fed with complex numbers the resulting graph forms waves.
Bernhard Riemann was a young German mathematician at work in the universities of Berlin and Göttingen in the mid 19th century. Europe in the 19th century was a fruitful place for mathematicians, with much crossfertilization of ideas taking place, particularly between the great mathematical minds of France and Germany. Riemann was especially excited by the emerging field of mathematics that combined exponential functions and complex numbers. He began to focus on the same area in his own research, feeding complex numbers into an exponential function called the zeta function and plotting the results on a graph.
A function in mathematics is like a computer programme in which you input a number, a calculation is made and another number is output. The following equation provides the rule for calculating the value of the zeta function when it is fed with a number x. To calculate the output it is necessary to carry out three steps. Firstly, calculate the numbers 1x, 2x, 3x,…nx Secondly, take the reciprocals of all the numbers produced in the first step (the reciprocal of 2x is 1/2x.) Thirdly, add together all the answers from the second step. The zeta function is symbolized algebraically by the Greek letter .
(x) = 1/1x + 1/2x + 1/3x + …. 1/nx + ….
As human beings, we live in a three-dimensional world. We orientate ourselves by means of three planes, the frontal plane which divides space into front and back, the sagittal plane that divides space into right and left, and the transverse plane which gives us our sense of up and down. When Riemann began to plot the equations of the zeta function on a graph the geometry that emerged was four-dimensional. It is virtually impossible for us to visualize this because our mental faculties are all based on the same three dimensional plan we inhabit physically, so the best we can do is to get an impression of it. The diagram below is a two-dimensional attempt to represent a four-dimensional world in a three-dimensional image! It is worth reminding oneself that it depicts a geometrical world that has arisen from the interaction of real numbers, with their infinite sequences of positive and negative whole numbers, rationals and irrationals, and the square root of –1.
Riemann used the complex plane to plot the numbers he input into the zeta function and the space above it to plot the output. If we can imagine the complex plane spread out flat on a table like a map and use threedimensional terminology, the results of each calculation is recorded ‘in the air’, so to speak, ‘above’ the input on the complex plane. The graph that emerged was composed of undulating curves, the sine waves typical of blending exponential functions and complex numbers. In his excellent book The Music of the Primes Marcus du Sautoy describes Riemann’s study of this geometry as though he was an explorer trying to pick his way through an uncharted landscape.
“As Riemann began to explore this landscape, he came across several key features. Standing in the landscape and looking towards the east, the zeta landscape levelled out to a smooth plane I unit above sea level. If Riemann turned and walked west, he saw a ridge of undulating hills running from north to south. The peaks of these hills were all located above the line that crossed the east – west axis through the number 1. Above the intersection at the number 1 there was a towering peak which climbed into the heavens. It was, in fact, infinitely high. Heading north or south from this infinite peak, Riemann encountered other peaks. None of these peaks, however, were infinitely high. The first peak occurred at just under 10 steps north at the imaginary number 1 + (9.986…) and it was only 1.4 units high.”
In the diagram the landscape stops abruptly at the north – south ridge of peaks. No landscape had formed to the west of these. In time Riemann succeeded in finding a new formula for the zeta function that enabled him to complete the landscape and he then had a map that encompassed all possible complex numbers.
Since Riemann died in 1856 generations of mathematicians have pored over his notebooks. Much of his work was destroyed in a fire, courtesy of a zealous housekeeper, but the surviving papers contain page after page of calculations that record his progress as he worked his way through the zeta landscape. The analogy with mountains and plains is particularly apt given the nature of the mathematics involved. Du Sautoy describes this as ‘rigid’. In mathematical terms this means (a) that it was not possible to make calculations with one set of variables without it affecting others and (b) the coordinates of one part of the map contained information relevant to other parts. Of particular significance were the points where the landscape fell to ‘sea level’. At intervals the zeta function output zero and Riemann discovered that the corresponding points on the map contained all the information required to construct the entire landscape. To understand the relevance of this from the point of view of the primes we need to return briefly to the fundamentals of number theory, and look at a different expression of the zeta function.
The prime numbers are the building blocks of number. Every number that is not a prime can be constructed by multiplying primes. For example, 15 is the product of 5 and 3, and 55 the product of 5 and 11. The same principle holds true no matter how great the numbers become. So 113,297 can be written as the product of the prime numbers 89 x 19 x 67. Mathematicians have devoted countless hours trying to find formulas for establishing whether numbers are prime, which is particularly important when it comes to large numbers, and for breaking odd and even numbers down to their prime constituents.
A century before Riemann, the Swiss mathematician Leonhard Euler had used the prime numbers as building blocks to re-write the zeta function thus:
(x) = 1/1x + 1/2x + 1/3x + 1/4x + 1/nx + …
= [ 1 + 1/2x + 1/4x + 1/8x … ] x [ 1 + 1/3x + 1/9x + 1/27x … ] x ..
… x [ 1 + 1/px + 1/(p2)x + 1/(p3)x .… ] x …
In this formulation, which is known today as the Euler Product, the zeta function appears on one side of the equation with reciprocals based on the harmonic series (½, 1/3, ¼…) and on the other side the reciprocals arise from each of the prime numbers in turn (the symbol p represents prime) and their sum is multiplied with those based on the next prime. Euler was another giant in the mathematical world and Riemann would have been familiar with this formulation. As he contemplated the zeta landscape Riemann realized that if the whole topography could be constructed from the zeroes, and from the zeta function when it was broken down into its primes constituents, there must be a connection between them.
Up until this time solving the puzzles of prime numbers had not been at the forefront of Riemann’s mind. He was encouraged to look more closely at the primes by one of his lecturers, Peter Lejeune-Dirichlet, who was one of the many mathematicians moving between the French and German universities. Also in Göttingen at the time was the legendary Carl Gauss. As Riemann began to focus on the primes his work began to rub shoulders, mathematically speaking, with Gauss’s own. Gauss was an old man by now and a virtual recluse. He lived just long enough to read Riemann’s thesis and appreciate his genius before dying in 1855.
The table below illustrates the rate of change in the proportion of primes from 10 to 10,000,000,000 through multiples of 10. It was Gauss who had found a way to estimate the number of primes to any number N and discovered that there was regularity in the rate of increase. It is in the right hand column of the table that this pattern becomes evident. Having varied slightly around 2 in the rate of increase, this settles down from 10,000 onwards closer to 2.3. Gauss had achieved these estimations through a logarithmic function based on the irrational number e. All the tables below are reproduced from The Music of the Primes and are based on modern calculations. By his seventies Gauss had managed to calculate the number or primes up to 3,000,000. We have to remember that he did not have the aid of computers!
Historically, Gauss had made a significant shift in perspective in order to arrive at this view of the primes. Up until then mathematicians had focused on the details of the primes. Gauss stood back in order to see the bigger picture. The logarithms he used gave him rough estimates of the number of primes rather than precise answers but this was sufficient for him to see that there was a rhythm in their distribution. Subsequently he developed a new function, called the logarithmic integral, denoted by Li(N), which produced a more accurate estimation. I should add at this point that the word accurate’ and ‘estimate’ become a little confusing to the layman. Modern calculations show that for the number of primes up to 1016 (which is 10,000,000,000,000,000) Gauss’s ‘estimate’ based on Li(N) deviated from the correct value by one ten millionth of 1 per cent! It was this estimate that Riemann improved upon as a result of the information contained in the zeroes of the zeta landscape. The refinement occurred in two steps. This table illustrates the difference between Gauss’s estimates based in the logarithmic integral Li(N), and Riemann’s, after he had taken the first step, which involved developing a new formula R(N) using the coordinates of the zeroes.
The second step occurred when Riemann discovered that each of the zeroes could be transformed using the zeta function into its own particular sine wave. These waves varied in height and size according to their location in the landscape. His function R(N) had improved on Gauss’s estimations for the number of primes up to N. He subsequently discovered that if he calculated the height of the waves, and added to those former estimations the height of each wave above the number N, he would arrive at the exact number of primes. When it came to pinning down prime numbers the zeroes of the zeta landscape had proved to be the ultimate treasure trove.
Mention the name Bernhard Riemann to anyone interested in mathematics and they immediately associate it with the Riemann Hypothesis. Proving the Riemann Hypothesis was one of seven mathematical problems offered as a challenge to mathematicians in May 2000 to mark the new millennium, with a bounty of one million dollars apiece. Riemann published his work on the zeta landscape in the monthly notices of the Berlin Academy in 1859. Hidden amongst the pages of dense mathematics was another puzzle concerning the location of the zeros which subsequent generations of mathematicians have struggled to understand. The function R(N) and its refinements had confirmed Gauss’s work and established without doubt that there was a pattern underlying the distribution of prime numbers. The Riemann Hypothesis concerns an even more mysterious order.
Evidently, establishing ‘location’ in four-dimensional terms is mathematically challenging! At some point in his research Riemann turned his attention back to focus again on the coordinates of the zeroes, despite the fact that he had already been studying them for years and using them in calculations. When he investigated the zeroes more closely he discovered that the first one was located at ½ on the east/west axis and 14.134725 on the north/south axis. The second was located at ½ on the east/west axis and 21.022040 on the north/south axis, and the third at ½ on the east/west axis and 25.010856 north/south. It has taken mathematicians many years to find out how he arrived at these coordinates. His belief that every single zero in the zeta landscape would line up on 0.5 is what has become known as the Riemann Hypothesis.
I said at the beginning of this paper that the story of Riemann’s work on prime numbers had raised fundamental questions for me. None of these questions have been answered, and judging by the response of my more mathematically-minded friends who I have badgered with questions, they never will be. In its most abstract form mathematics seems to operate in the realm of paradox and come up against the limits of human conceptual ability. Despite this I would like to end by trying to formulate the questions, or, at least, their ‘ball park area’. I ask the mathematically-minded to excuse my naivety.
To you and I, counting our way through the whole numbers, claps on the primes seems to be random. We clap 3 times (there is some dispute as to whether 1 is a prime number but I am assuming that it is) and then pause for 1 beat, clap on 5, pause for another beat and clap on 7, pause for 3 more beats and clap on 11. If we were patient enough to continue for some time we would find ourselves clapping on 83 and then pausing for 6 beats, clapping on 89 and then pausing for 8 beats before clapping on 97. And if we were extremely patient we would clap on 9,999,971, pause for 1 beat and clap on 9,999,973 and then pause again for 18 beats until we got to 9,999,991. This is pretty childish stuff compared to the sophisticated calculations that mathematicians deal with, but we have something in common with Carl Gauss. Gauss’s logarithmic integral was based on real numbers. We are at least looking at the primes through the same lens.
Mathematics is concerned with order. Mathematicians seem to be primarily motivated by delight in order, and order is what they require of mathematical proof, which has to be so rigorous and precise in its logic that it is absolutely unassailable. Through the lens of the real numbers the primes seem to behave in a pretty chaotic fashion. Through the lens of the complex numbers, the zeroes, which are so bound up with the distribution of the primes, are strictly regimented like a line of soldiers. The lens through which Riemann looked at the primes was based on the square root of minus 1. The square root of minus 1 cannot exist from the point of view of the conventional rules that govern positive and negative numbers. What does it say about our view of the world when mathematicians, whose governance structure is based on rigorous principles, discovered an ordered system emerging from an apparently contradictory concept?
In The Music of the Primes there is an extraordinary sentence. Du Sautoy is describing work on the Riemann Hypothesis conducted by the British mathematicians G.H. Hardy and J.E. Littlewood, who formed a famous mathematical double-act during the first half of the twentieth century. Referring to Hardy’s study of the zeroes on 0.5, he writes
“Hardy had taken the great step of proving that there were at least an infinite number of zeros on the line, but he had failed to show that this infinite number amounted to even a fraction of the total number of zeros.”
There are, in fact, many passages in Du Sautoy’s book that describe extraordinary phenomenon, but this particular one stopped me in my tracks. Infinity cannot have totality, so how can an infinite number of something be a fraction of its total amount?
When talking about numbers and amounts we seem to be in strange territory. We know that there are an infinite number of integers (whole counting numbers and their negatives) and consequently that there are an infinite number of rationals. The German mathematician George Cantor established in 1873 that if all the whole numbers are compared with fractions they pair off exactly. This is not the case with irrational numbers. Apparently, these slithery creatures form a different order of infinity. This seems to contradict the notion of infinity. If something is infinite surely it is boundless? And if it is boundless how can it be different? Are we saying that infinity itself is a rational number?
In the history of the search for order, in prime numbers zero has celebrity status. The complex plane pivots on zero, and it is the zeroes in the zeta landscape that produce the key that unlocks their secrets. Are we talking here about the same zero or different versions? Are each of the zeroes in the zeta landscape the pivotal point of the complex plane? And is it even possible to ask what zero is, let alone divide into fractions or versions? Of the two axes that form the complex plane, one represents the real numbers and the other the imaginaries, to one side positive and the other negative. Does this mean that zero is not real or imaginary, positive or negative, or does it partake of all of these?
The Music of the Primes – Marcus du Sautoy
The Art of the Infinite – Robert and Ellen Kaplan
http://mathworld.wolfram.com/RiemannZetaFunction.html for more serious mathematics.