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Explore the Back-StoryAuthor: Kazi Abu Rousan

Where are the zeros of zeta of s?

G.F.B. Riemann has made a good guess;

They're all on the critical line, saith he,

And their density's one over 2 p log t.

Source https://www.physicsforums.com/threads/a-poem-on-the-zeta-function.16280/

If you are a person who loves to read maths related stuff then sure you have came across the words **Riemann Zeta function** and **Riemann Hypothesis** at least once. But today we are not going into the **Riemann Hypothesis**, rather we are going into the **Zeta Function**. To be more specific, we will see how to calculate the value of **zeta function** using a simple **Julia Program** only for $Re(input)>1$.

What is the **Zeta Function**?

The Riemann zeta function $\zeta (s)$ is a function of a complex variables $z = \sigma + i t $. When $Re(z) = \sigma >1$, we can define this as a converging summation given by,

$$ \zeta(z) = \sum_{n = 1}^{\infty} \frac{1}{n^z} = \frac{1}{1^z} + \frac{1}{2^z} + \frac{1}{3^z} +\cdots $$

This definition is so simple right?, Here $z$ is a complex number. If it is taken as real, then we will get many famous series like,

**Harmonic Series**(Diverge as $Re(z)=1$): $H = \zeta(1) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$

**Basel problem Series**(Converges): $ \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}$

and many more.

There are many methods to calculate the values of this function for each value of $n$. You can surely do this by hand although I can't suggest that. The easiest way will be to utilize program.

Before writing the code, you should remember that the **sum is actually infinit**e. Hence, our function will give us more and more fine result as we include more and more terms.

```
function ζ(z,limit=1000)
result = 0
for i in 1:limit
result += (1/i)^z
end
return result
end
```

This is the function. How simple!!.. and If we use **julia** as you can see, we can actually use $\zeta $ symbol. Pretty cool right?

```
ζ(2,10_0000_000)
#Output: 1.644934057834575
#where pi^2/6 = 1.6449340668482264
#pretty close
```

It's not all. We can actually apply this function to find the value of zeta function for complex inputs too.

```
ζ(2+5im)
#Output: 0.8510045028264933 + 0.09880538410302253im
```

When we use complex inputs, there is a beautiful hidden beauty. Let's see that using a plotting library called Plots and we also have to rewrite our function a little bit.

```
function ζ(z,limit=1000;point_ar = false)
points = ComplexF64[0]
result = 0
for i in 1:limit
result += (1/i)^z
if point_ar
push!(points,result)
end
end
return result, points
end
```

Now, we can use this to plot.

```
z = 2+5im
result, points = ζ(z;point_ar = true)
plot((points),color=:blue,width=3, title="Zeta function spiral",framestyle= :origin,label="$z")
scatter!((points),color=:red, label="Points")S
```

The output is:

Who would have thought that there will be something like this hidden.

Now, If we apply this function to all possible points of the NumberPlane, then we will get a mesmerizing pattern.

Looking at the image it feels like It's begging to be extended to the other portion. This extension is done using something we called * Analytic Continuation*. We will not go into much detail here.

If you want to know about the remaining story visit by lecture here:

This is all for today. Why not try to extend the idea to the other side?

Hope you learnt something new.

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