Sue Jiang
Music of the Primes Chapter 4
Riemann made a breakthrough in mathematics by combining the zeta function
with imaginary numbers, something that has never been thought of before. His ideas are
what opened up a whole new outlook on the primes.
In Berlin, Riemann worked with Gauss on his extensive research of the zeta
function. Gauss admits that what really impressed him about Riemann was his strong
geometric perception about feeding functions with imaginary numbers. To Riemann, the
equation of a function is only the starting point. It is the geometry of the graph that is
what is most significant.
The most obvious problem with working with imaginary numbers is that they
cannot be graphed. Thus, Riemann used four dimensions in order to illustrate his ideas.
Most mathematicians think of the fourth dimension as time, but in reality, it can be
whatever we are interested in tracking. The author compares this to a graph an economist
draws when comparing four things: interest rates, inflation, unemployment, and national
debt. Although the graph of economy itself cannot be drawn, it is the topography of the
graph that reveals the relationship between these four things.
Unfortunately, most people cannot possibly imagine what a four-dimensional
graph would look like. Thus, we create a three-dimensional shadow that provides
sufficient information to analyze Riemann’s ideas. This shadow is similar to the 2D
shadow of a 3D object. There is one obvious drawback to using the shadow to analyze
the graph, which is that a significant amount of information is inevitably lost, just as a 2D
shadow cannot fully describe a 3D object.
Gauss created a two-dimensional map of the imaginary numbers. The north-south
axis shows the steps in the imaginary direction and the east-west axis keeps track of the
real numbers. The height of the graph is the output of the function.
When Riemann examined the 3D shadow of the zeta function, he saw that when
looking towards the east, the landscape smoothed into a plane that is one unit above sea
level. However, turning around a looking west, he saw undulating hills running up and
down the north-south axis. At the number one, the graph spiraled off into infinity.
Riemann also observed that when he inputted any number to the west of 1, the result
always spiraled off into infinity. Thus, it seems as if the graph ended at the number 1.
Riemann was baffled by this observation, and soon created another formula that
completed the graph to the west. This new formula formed a seamless border between the
east and west.
When observing the new graph he created, Riemann saw that the geography was
extremely rigid. Riemann could not change the landscape or formula created; otherwise
the seam between the landscape would tear. From this observation, Riemann deduced that
he could use any small region of the imaginary landscape to construct what the entire
graph would look like. This is counterintuitive because one would not expect to be able to
construct an entire mountain range just by observing a few hills. Riemann’s second
discovery regarded the zeros of the zeta function—where the graph was at sea level. He
found that by only calculating the zeros of the function, he is able to construct the entire
graph.
Next, Riemann related Euler’s product and the zeta function. He found that the
prime numbers and the zeros of the zeta function built the same landscape; this could not
have been a coincidence. A couple of years ago, Gauss attempted to find a formula that
would exactly calculate the number of prime numbers up to a number N, but failed.
Riemann was soon able to solve Gauss’s enigma using the zeros of the zeta function and
imaginary numbers. The imaginary numbers provided Riemann with a method to undo
the errors in his calculations. Riemann also observed that feeding an imaginary number
into the exponential function produced a sine wave. He realized that all of the zeros of the
imaginary landscape could be transformed into its own special wave.
Riemann’s original formula for counting the number of primes up to a number N
was R(N). This formula gave a reasonably good count, but was definitely not precise. But
by adding to this formula the height of each wave above the number N, he could find the
exact number of primes. Thus, it was the waves created by the zeros of the imaginary
function that unlocked the secret behind the primes. When comparing Riemann’s new
function with the stair-case function of the prime numbers, they are almost identical. But
his original function, R(N), was smooth and did not look anything like the staircase
function. It is only after adding the waves of the zeros to the function that the function
begins to look more like the staircase graph.
Riemann also found that the sine graphs created by the zeros of the zeta function
represents sound waves. Thus, Riemann found the music behind the primes. The sine-like
waves that Riemann had created from the zeros in his zeta landscape revealed the
harmonic structure of prime numbers. Riemann’s imaginary world threw together simple
structures, such as sine waves, to produce the music of the primes. When played, this
sound was the music of the primes. What would this sound like? One might expect this
harmony to be white noise since there is an infinite number of sine waves, but the sounds
are actually single clear notes with no harmonics.
Another mathematician whose discoveries greatly influenced Riemann’s works
was Fourier. By the age of 13, Fourier was entranced by the world of mathematics.
Unfortunately, Fourier was destined to become a monk. Fortunately, he was later freed
from his responsibilities and was able to pursue his passion for mathematics.
Napoleon was also impressed by Fourier’s ability as a mathematician and teacher.
Napoleon took Fourier on many of his sea expeditions. Fourier’s main job was to
entertain Napoleon with intellectual inquiries, such as the age of Earth and whether or not
other planets were inhabited. On one of his voyages to Egypt, Napoleon abandoned
Fourier, who lived in Egypt for quite a while before returning to France. Back in France,
Fourier developed a strange affinity for heat. Much of his academic work was dedicated
to describing how heat propagated through matter. His ideas were much acclaimed, but
also received a lot of criticism at the Grand Prix in Paris. The judges thought his treatise
contained mistakes and a lot of his mathematical explanation was not thorough.
Next, Fourier made many graphs of heat and temperature and found they these
graphs represented sound waves. The horizontal axis represented time and the vertical
axis represents the volume and pitch of the sound at each instant. Fourier explored how
sounds and harmonies are represented as sine waves of different frequencies. He found
that the reason instruments sounds so different even when playing the same note is
because the shape of their graphs are different. Thus, the intricate and complex sound of
an orchestra and be broken down into the sum of simple sine waves. However, sine
waves can sum up to produce white noise as well, which is a mix of an infinite number of
sine waves.
Next, Riemann tried to find a zero that created a sound wave that was louder than
the others. He started testing the zeros, but disregarded the ones to the west because they
had no pitch and were called trivial zeros. When Riemann actually calculated the location
of these zeros, he found that they all lay on the same east-west point: ½. He knew this
could not be a coincidence, which led him to propose the Riemann Hypothesis. This
states that all of the zeros lie on this critical line. Riemann had now created order to the
random and disorderly world of prime numbers. He knew that this critical line was an
important line of symmetry to order the zeros.
Despite this breakthrough in mathematics, Riemann did not publish much of his
discoveries. Instead, he remained modest and admits that he did not try too hard to prove
his hypothesis. In fact, his main goal was to prove Gauss’s prime number conjecture: to
show why Gauss’s guess for the primes became more accurate the more primes one
counted. After publishing his paper, Riemann led a content life for a couple of years.
Unfortunately, in 1862, he was plagued by chronic ill health. His later life was not
especially joyful, and he died at the age of 39. After his death, his housekeeper and
widow lost many of his papers, some of these probably including profound and
significant ideas. Frustratingly, Riemann probably proved much more than he published