One of the more esoteric and mystical explanations of prime numbers is an unproven hypothesis called the Riemann Hypothesis. It involves the use of prime numbers and the zeta function, which when taken to an “imaginary” and “complex” realm, expose a deeper meaning to the foundation of number theory. Before we dive in, we’ll pay our due diligence covering what primes are and work our way up through the millennia of mathematical work that laid the groundwork for Riemann to hypothesize the placement of prime numbers along a unique line, across an even more unique graph.
To better get a handle on the Riemann Hypothesis, we’re going to cover a few seemingly-disparate concepts in chunks and then tie them together towards the end. We’ll be covering:
- Prime numbers & the prime counting function → why numbers that are indivisible are so important, infinite, but elusive
- Convergent & divergent series and the zeta function → why functions that add numbers in an infinite series can have a finite or infinite answer and why the zeta function in particular is most interesting
- Euler’s product formula → how the zeta function, an additive series, is really a product series with prime numbers in disguise
- Imaginary & complex numbers → why defying math by creating a whole set of contradictory numbers unearths a hidden mathematical reality
- Plotting the Riemann zeta function on a graph→ placed on a graph, a single number plays a mystifying role in the building blocks of numbers
- Plotting the Riemann zeta function on polar coordinates → using polar coordinates, prime numbers create a beautiful pattern
Alright, now let’s dive in!
If we take a series of numbers and add them on a line, we get a real number line. We can extend this on to infinity because every number we increment, can always be followed by another. In fact, we can extend numbers the other way as well and decrement them until we enter the realm of negative numbers -- the “anti-number” of sorts that can have a cancelling effect when paired in conjunction with another number (i.e. summing 5 with -1 gets you 4).
It’s a great system -- a number-generating system. On that real number line we can create new numbers by combining them together (addition and subtraction), compounding them (multiplication), or reducing them in chunks (division). It’s this last one, division, that really pares a number down to its most basic bit. I can divide 6 by 2 and get 3 -- a new number! I can also divide 6 by 4 and get 1.5 (meh, not really a number but a pseudo number, still seen as a combination of two numbers: 6 in 4 parts, or 3 in 2 parts). Getting crisp clean numbers, integers, is an exercise that pares down the essential bits of numbers that can be used as building blocks to build other numbers: queue in prime numbers.
A prime number is an integer that has really been broken down to its most essential bit -- it can only be divided by the number 1 and by itself. These are known as its divisors.
Prime numbers are the basic building blocks of all integers, which leads us to the Goldbach Conjecture. Christian Goldbach (1690-1764) stated that every even integer greater than 2 can be written as the sum of 2 primes. What validates this theory is the notion that there are an infinite number of primes, an idea shown by Euclid 2,300 years ago using an exhaustive method known as the Sieve of Eratosthenes. With the Sieve of Eratosthenes, you can start with any group of numbers -- we’ll start with 1 to 100, and begin to cross out all numbers that are divisible by a number. We cross out 1 because it is not prime, and begin with the number 2. All even numbers are divisible by 2, so we cross them out. Next, we take the number 3 and cross out all remaining numbers that are divisible by 3. Then we progress to multiples of 4, 5, 6, 7 and so on until we have prime numbers remaining. This method works in shaking out all the primes in a group of numbers -- it doesn’t prove that there are an infinite amount of numbers, but you can take large chunks of numbers far down the real number line and perform this exercise to see that you’ll always find primes.
With these infinite numbers of primes, can we find a pattern to their distribution? At first glance, when looking at the distribution of prime numbers, we can deduce not a pattern, but a common occurrence. Occasionally, when we find a prime, we find another prime number two digits down. The distribution of these prime numbers with a 2 digit gap, isn’t really a discernable pattern that we can deterministically define -- it’s a little chaotic but not random. Known as the twin primes conjecture, it states that there are an infinite number of these pairs, called twin primes. This conjecture has yet to be proven, but with tools like prime k-tuples, we’ve gotten closer to decoding their pattern.
In 2013, a Chinese mathematician named Yitang Zhang released a paper that didn’t solve the twin primes conjecture, but showed a novel way in which one could prove that a gap amongst primes along the infinitely long real number line can be anywhere between 2 digits to seventy million. While seventy million seems like a large number (definitely much bigger than 2), it is still small compared to infinity. What was important was his method, which set the tone for mathematicians to continue that thinking to narrow the gap between 2 and seventy million. Sequential proofs were submitted online and as of the writing of this paper, we’ve closed the gap to as small as 12. The idea is that using this tool Zhang created, we can theoretically lower this limit to 12, but not 6.
Separate from the pattern, we find that primes become sparsely distributed the higher we increment our values. You see, when we count the number of primes under 100, we get 25 primes -- 25% of all integers. When we count the number of primes from 1-1,000, we get 168 primes -- 16.8%. When we count the number of primes from 1 to 10,000, we get 1,229 primes -- 12.2%. In fact, when we count the number of primes from 1 to 10 Million, we get 664,579 -- 6.6%. The more we increment our numbers, the less primes that occur. This gradual reduction of its occurrence is asymptotic and we’ll get to why this is important. As of now, just remember that even though the distribution of primes thins out, it doesn’t stop.
Now that we’ve become familiar with primes and the importance in our understanding them better, we’ll explain another important concept that leads us to the Riemann Hypothesis: convergent and divergent series.
Earlier we showed a real number line with the divisors of each integer to demonstrate what a prime number is. Now, we’ll be using something called a Prime Counting Function to get a visual of this asymptote.
Carl Friedrich Gauss used a graph whose x-axis are real numbers and whose y-axis are the number of primes. This Prime Counting Function would have us move along the x-axis and increment vertically a plot for a prime number we encounter. The graph looks like the one below -- conveying the asymptotic growth we alluded to before.
Prime Counting Function
At first glance, this looks like logarithmic growth -- a logarithm is how we undo exponentiation the same way division undoes multiplication. Specifically, this function looks a lot like 1/log(x), though we can’t quite get the two to overlap perfectly -- there is still some jaggedness to our Prime Counting Function.
Still, this brings us closer to uncovering the secrets of prime numbers and their distribution. It’s known as the Gauss Conjecture and in arriving at this insight, Gauss determined all prime numbers up to 3 Million for his inputs -- not bad for the 19th Century which predated calculators. We’ve since distilled this to what we call the prime number theorem, where the number of primes up to x is about x/log(x).
For one thing, we know that these prime numbers extend infinitely, even as the distribution of primes are reduced the farther we increment. This is known as a divergent series, in which the succession of numbers extend infinitely and the sum of those numbers in the series have no limit, hence there being an infinite number of primes. This is the antithesis of a convergent series, which also extends infinitely but converges towards a specific number.
Let’s take an example of a divergent series as shown above. With every computation in this function, we notice the result becoming larger and larger -- we can infer that the solution is infinite.
/quality=80,fit=scale-down)
Now, for another example of a divergent series as shown above.. This is known as the harmonic series, and with every number you increment, the value increments, but at a slower rate. Surprisingly, it seems like we’d approach a single value where we’d eventually stop -- but because of our infinite ability to add to the series, there will always be a new value as the output.
An example of the convergent series is shown above, which converges to the number 1. The more numbers we add to the series, the closer (but slower) it approaches 1. The speed in which it approaches the number is where we can place our confidence that that is indeed the number it will converge to.
Series like these produce beautiful yet unexpected results and so Leonhard Euler in 1737 produced what he named the zeta function. The zeta function is shown below -- where zeta of s is equivalent to the harmonic series below.
By applying different positive integers for s in the zeta function, we find very interesting results:
- When s(-1) = -1/12
- When s(0) = -1/2
- When s(1) = Infinity
- When s(2) = pi^2/6
- When s(3) = 1.20205690...
- When s(4) = pi^4/90
Great, so we found unique properties in our infinite series, particularly the zeta function, but what does this have to do with prime numbers? Euler drew a connection between the zeta function and prime numbers using a function he devised called the Euler Product Formula.
In it, he was able to show that the zeta function was the same as his product formula which is the multiplying of the inverse of a prime number to the negative power of any given integer -- similar to the zeta function. They are both harmonic series -- but how does a harmonic series of adding the inverse of every positive integer to a given power the same as multiplying the inverse of every prime number to a given negative power? It still shocks me that in the 18th Century, Euler knew to even attempt the two and find a bond between them. Because this series diverges, it can be implied that there are an infinite number of primes -- although that has yet to be proven.
Now, we can make our way to the Riemann Hypothesis. Fundamentals of cryptography and quantum physics rely strongly on the Riemann Hypothesis being true and have advanced under the assumption that it is true -- but this hypothesis has yet to be proven formally.
Riemann took the zeta function and asked an interesting question: What would happen if you allowed the zeta function to take complex inputs?
A quick primer on complex numbers: earlier we saw that we can represent a number in many ways, as a radical, decimal, fraction. We can take all of these numbers and plot them on the real number line because theoretically they should exist. Even if we had to squint and really shrink ourselves between integers, we always found where things should exist on the real number line.
If you were to walk backwards on the number line and find negative numbers, it’s not difficult to imagine why a negative radical number should exist -- it’s merely a different way of representing a number. This is what we call an imaginary number, one whose square is a negative value. This is contrary to what we’ve learned about in multiplication. Multiplying any two combinations of positive or negative numbers should not yield a negative number, but alas here it lies on our number line. The mathematician Gottfried Leibniz called imaginary numbers “a wonderful flight of God’s Spirit; they are almost an amphibian between being and not being.”
We represent imaginary numbers with the letter i. The term was coined by Leonhard Euler after the Latin word imaginiarius but the discovery of the imaginary number stemmed farther back. Ganita Sara Samgraha, a book written in 850 A.D. by the Jain scholar Mahavira, was the oldest documented source that explicitly stated the square root of a number did not exist.
It wasn’t until the sixteenth century that the imaginary number was accepted as a type of number used in calculations. Granted it wasn’t widely accepted, Renee Descartes was still averse to their use (remember, he doubted everything other than his thinking self and built up from there, “Cogito Ergo Sum”, so his pragmatism can be understood). Now, where Descartes and imaginary numbers cross paths is when we add a perpendicular line through the real number line, a y-axis that we use to now create a complex plane.
If we represent real numbers on the real number line (x-axis) and show the imaginary numbers on the perpendicular line (y-axis), then we can imagine using a combination of the two to plot a number that’s a combination of the two. This is known as the complex number, as it’s a combination of a real number and an imaginary one. In a way, we made these numbers up, yet we use imaginary and complex numbers all the time in our math which are applied to the sciences in real life -- it’s used in alternating current electronics.
And so, this is also used in the zeta function, specifically now known as the Riemann zeta function. Taking those complex inputs is like asking what it would be like to extrapolate this harmonic series to the edge of reality. This leads us to what exactly the Riemann Hypothesis is: a conjecture stating that the Riemann zeta function has its trivial zeros only when the real numbers are negative even integers and has non-trivial zeros at the complex numbers with real part ½. It took us a while to get to making this statement -- mainly because all the background context was needed, but we’re still not finished.
Trivial and non-trivial zeros are an important differentiator here. As you can see in the diagram below, all the trivial zeros fall on all negative even integers. This is an interesting pattern, but the placement of our “zeta zeros” on the complex plane (when y is a non-zero value) is of utmost interest to us. All non-critical zeros fall within what we call the critical strip, which is between 0 and 1. Riemann stated that there are an infinite number of zeta zeros found on that critical strip; they fall within the critical line when s = ½. If you add up all the harmonics of the zeta zeroes, you get a perfect match to Gauss’s modified prime counting function.
With the Riemann Zeta Function, when s = 1 the sum is infinite, but when s is greater than 1 the sum is finite. Euler’s work was used to show that the rate at which Z(s) explodes as s approaches 1 can give you information about how often primes occur.
If you’re not convinced that the distribution of these zeta zeros on the critical strip says something about prime numbers, we’ll show an example using polar coordinates.In our 2-dimensional Cartesian plane, a placement on the graph can be indicated using scalars, or coordinates following the x and y axes. Each point can be traversed either vertically or horizontally. On a polar coordinate system, each point is indicated on the plane with respect to a distance from the reference point and an angle from the reference direction.
The Riemann zeta function when charted on a polar graph creates a beautiful and complex looking curve that is used in number theory to investigate the properties of prime numbers. However you decide to plot the Riemann Zeta Function, it hints at deeper meaning behind the prime number and gets us closer to understanding why they are the basic building blocks of all numbers.