But hey, maybe you’ll find it fascinating!

So what the ***** is an elliptic curve? Well, let me put it this way: imagine you have a piece of paper and you draw two points on it. Now connect those points with a line that doesn’t touch any other point on the paper (except for where the line starts or ends). That line is called a “line segment.”

Now take another piece of paper, and this time draw three points on it. Connect each pair of points with a line segment, but make sure those lines don’t intersect anywhere except at the endpoints. This creates what we call an “elliptic curve,” because if you add up all the angles between these lines (called “line segments”), they will always add up to 360 degrees just like the angles around a circle, or an ellipse!

Okay, so that’s kind of cool. But why do we care about elliptic curves? Well, it turns out that they have some pretty interesting properties when you start doing math with them. For example, if you add up two points on an elliptic curve (called “point addition”), and then repeat this process over and over again, you’ll eventually get back to the starting point! This is called a “periodic” property, because it repeats itself in a cycle.

But wait there’s more! If we take that same elliptic curve and multiply one of its points by an integer (called “point multiplication”), we can actually get back to the starting point after doing this multiple times as well! This is called a “finite” property, because it only takes a finite number of steps to get from any given point on the curve to another.

So what’s so special about these properties? Well, they have some pretty cool applications in cryptography (the science of keeping secrets). For example, if we use elliptic curves to encrypt and decrypt messages, it becomes very difficult for anyone else to read those messages without the proper key! This is because the math involved in point addition and multiplication on an elliptic curve is incredibly complex so much so that even a supercomputer would have trouble breaking the code.

But enough about cryptography zeta functions! A “zeta function” is basically just a fancy way of saying “sum.” For example, if we add up all the numbers from 1 to 10 (called “the sum of the first ten natural numbers”), we get:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

That’s a pretty simple zeta function. But what if we wanted to add up all the numbers from 1 to infinity? Well, that would be impossible there are just too many of them! However, mathematicians have come up with some clever ways to “sum” these infinite series using something called an “integral.”

An integral is basically a fancy way of saying “area under a curve.” For example, if we want to find the area between 0 and 1 on the x-axis (called “the interval from 0 to 1”), we can use calculus to calculate that value. But what if we wanted to find the area between -infinity and infinity? Well, that would be impossible too!

However, mathematicians have come up with some clever ways to “integrate” these infinite series using something called a “zeta function.” A zeta function is basically just a fancy way of saying “sum,” but instead of adding up numbers like we did before, we’re now adding up the reciprocals of those numbers.

For example, if we wanted to find the sum of all the reciprocals of the natural numbers (called “the harmonic series”), we would get:

1/1 + 1/2 + 1/3 + 1/4 + … = ???

This is an infinite series there are just too many terms to add up! However, mathematicians have come up with some clever ways to “sum” these infinite series using something called a zeta function. By doing so, we can calculate the value of this sum (called “the Riemann zeta function”) and use it to study all sorts of interesting properties in math and physics!

Who knew?