Riemann Zeta Function

It is a function, which has the variable (zeta), which is the summation of all of the fractions (s), where s is the number of numbers.

Now you may ask.

Why is this important?

It can lead to very beautiful results.

For example, let us take zeta(2)

The result turns out to be (pi^2/6)

For other instance let us take zeta(4)

The result turns out to be (pi^2/90)

The first one is also called the Basel's problem.

It can easily be solved using Euler's method.

1. Expand the Taylor series, and then recall it for the sine function.

2. lim (x-> 0) sinx/x = (1-x/pi)(1+x/pi)(1-x/2pi)(1+x/2pi)......(1-x/npi)(1+x/npi), for some natural number "n".

3. (a+b)(a-b) = a^2-b^2. Solve it again by the method.

4. It equals zeta(2) as all are two degree fractions.

5. Hence zeta (2) = pi^2/6.

In addition its also quite beautiful due to other reason.

It opens the door to the Riemann Hypothesis.

Let say, we take s = 1 + i

i = iota (or) we call it the imaginary number

Now the graph will make looks on the function:

Now 1/1+i = 1+1^-i.

1 is the magnitude.

(-i) is the amount of rotation done on the complex plane.

Now the entire graph only exists on the left side of the plane.

If mirrored, we can see that the angles remain the same, that means it is a differentiable function.

Thus, this is the basics of Riemann Hypothesis covered,

Thank you for reading! :)