The Riemann Zeta Function

It’s easy to believe that everything can be divided into clear cut categories, juxtaposed against each other, creating beautiful symmetry. To believe this, is the same as believing life is fair. It isn’t. And it never will be.

Mathematics isn’t the mass murderer of creativity it’s made out to be. In fact, it takes an artist to see its true form.


Riemann? What is that?

It’s not a what, it’s a who. Bernhard Riemann was a German Mathematician who made Herculean contributions to Mathematics, even if he did leave us with his mind-boggling hypothesis.

Zeta? What is that…greek?

Sure is. It’s a greek letter. It’s used as a unique symbol in The Riemann Hypothesis to represent the equation of interest.

Function? I feel like I remember that from math class.

You do. It’s an equation that relates an input to an output. Put in revenue and get out profit, or put in exercise and get out weight loss, it’s about creating a relationship between numbers.

The Riemann Zeta Function is an equation Bernhard Riemann developed and used the Greek letter Zeta to identify it. The input to the Riemann Zeta Function is the variable s, which we’ll talk about later. The full equation is shown below.

\zeta(s) = \sum\limits_{n=1}^\infty \frac{1}{n^s}

This equation starts with sigma (for the readers who could brush up on their greek letters – the abstract E)  which represents a sum of a number of mathematical terms based on the expression that follows it, in this case, the \frac{1}{n^s} . The number at the bottom of the sigma symbol tells us the starting number and the number on top tells us the number of terms to sum. For the Riemann Zeta Function, the output would be the total sum of an infinite number of terms, an example would be

\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...

all the way to infinity, which is seen as the symbol \infty. For the Zeta function, s is an exponent.


With all the facts and logic so far, I’m sure I’ve driven you away from the point I was trying to make…that Mathematics requires creativity. So a question for you,

How can you add up an infinite number of things?

If you can’t imagine how to do this, you need more imagination. Thanks for making my point!

Adding a never-ending stream of terms is a strange concept, but at times it’s possible. Let me give an example.

What do you see when you look at the equation below?

\sum\limits_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+...= \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...

It forms a pattern and we’re adding up the numbers in the pattern. Can you give the pattern a form? Could you create a drawing representing it? Could you sculpt something to describe it? Could you create music that sounds like it? This is where creativity comes in!

In this case, you can see that each term is 1/2 the size of the term before it. If you think of these terms as filling up a 1×1 square, as more terms are added, the entire square would be filled.

Screenshot 5:31:15 1:12 PM

This is only one form to imagine. I’d love to have people submit other forms that equation inspires in them. If you’re interested email me at infinit.brandon@gmail.com with a submission of your interpretation and a brief explanation, if I get enough unique submissions I’ll create a blog post showcasing them.

Since the entire area is 1, and each term added gets us closer to filling the entire square, as we approach an infinite number of terms, the value becomes 1. So the sum value can be written as follows:

\sum\limits_{n=1}^\infty \frac{1}{2^n} = 1

That example shows that infinite terms can sum to an actual value. I hope you find that interesting. If you don’t, then here is a fact, for some of these infinite sums that sum to finite values, a computer can never calculate them, it takes a person finding a form that is calculable. How’s that for countering the rise of the machines?

What is the Zeta Function doing?

Just as \sum\limits_{n=1}^\infty \frac{1}{2^n} can be given a form that can be visualized, the Riemann Zeta Function also has an interpretation. In fact, I’m sure there is an infinite number of interpretations and that’s the point. It’s people who can use their understanding of Mathematics and combine that with creativity who solve the unsolvable.

Proofs lacking creativity are pure exercises in Mathematics, strictly coherence to rules already established. Boring! Beyond boring, it leads us down long paths, not knowing where we’re heading but in a hurry to get there. Plus…no one wants to review them!

I want to explain what the Zeta Function is doing in an imaginative way, but a few details are still missing, like the variable, s, which I’ll cover soon. Until then, I’ll leave you with this, imagination is one way to come up with solutions to math problems, but more importantly it’s a way to find a way to state the problems themselves.

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