The Complex Variable s

Does something imaginary have no value?

If I tried to sell you an air guitar, you wouldn’t buy it. Unless I REALLY rocked it in a demonstration! Just Kidding.

Does that mean the air guitar has no value? The Air Guitar World Championships entertains tons of people, they even sell tickets to see it. This is just one example where something imaginary has no value itself, but value can be derived from it.


 

It isn’t the difficulty to understand that makes the variable, s, complex; it’s that it’s a combination of a real number and an imaginary number. If you don’t know what an imaginary number is click here.

In the Riemann Zeta Function, \zeta(s) = \sum\limits_{n=1}^\infty \frac{1}{n^s}, s is a variable that represents a complex number. It’a also the input to the Zeta Function. The variable s, as it relates to the Riemann Hypothesis is defined:

s = \alpha + \beta{i}

In this case, alpha (\alpha) is defined as being only a real number, which are numbers you know like 1,2,3.141598,-5, etc., while beta (\beta) times the imaginary number, i, represents the imaginary part of the complex number, or essentially a square root of a negative number that doesn’t have a physical interpretation, but arises often in physics.

Using s as a placeholder in the Zeta Function prevents having to write alpha and beta over and over again. The Riemann Hypothesis is a statement about alpha (\alpha). It says that under certain conditions, which we’ll talk about soon, the only way \zeta(s)=0 is if \alpha=\frac{1}{2}.

Time to visualize!

Imagine we have a race taking part on a street. The winner is the first person to run 1 mile down a straight path. This distance is measured in our alpha direction in the image below. Now imagine at the same time the races starts, an earthquake happens. For official reasons, the race has to continue, and can’t be reran. Due to the tremors, the contestants can’t seem to run straight anymore. As a result, they are veering off the straight-line and end up running more than a mile, but still the race is judged on the straight mile and not the wobbly one. We’ll call their fluctuations from the straight-line, our imaginary beta term. At the end of the race, runners were 1 mile from that start in the alpha direction, but were also some distance off the path, their actual position is similar to the variable s.

Running Path

So what’s the point of an imaginary number?

In this example, while only the time required to run the real distance, alpha matters for winning the race, it doesn’t necessarily mean that the winner was running at the fastest pace. For that we would need to know the length of the path each runner took divided by their finish time. If a faster runner had a longer path because of extra deviation from the road by the earthquake, then they might not win. Without tracking the fluctuation, beta in this case, it’s impossible to calculate the actual distance ran, or running speeds. Imaginary numbers are important because often times they are used to calculate other additional items.

To wrap up, the imaginary part of s, $\beta{i}&s=1$, produces valuable changes in the Zeta Function, but it in itself is not of value to the hypothesis statement which we’ll see in the next chapter. If you’re not clear, don’t worry. The variable s is just a definition describing real (\alpha) and imaginary (\beta{i}) terms.

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