First, I am not an expert mathematician at all, but I was just curious to explore and understand this topic a little better. I might had some mistakes along the way so if you find any, please point them out.
So, I started with the zeta function: $$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^s}$$
When $s=x+yi$
$$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^{x+yi}}=\sum_{n=0}^\infty \frac{1}{n^xn^{yi}}$$
Then I expended the $n^{yi}$ term:
$$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^xe^{iy\ln{n}}}$$
And by Euler's Identity:
$$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^x(\cos(y\ln{n}+i\sin(y\ln{n}))}=\sum_{n=0}^\infty \frac{1}{n^x\cos(y\ln{n})+in^x\sin(y\ln{n})}$$
Now it is a reciprocal of a complex number. I used $\frac{1}{u+vi}=\frac{u-vi}{u^2+v^2}$ to get a new complex number (call it $z=u+vi$):
$$\zeta(s)=\sum_{n=0}^\infty z_n=\sum_{n=0}^\infty u_n+v_ni$$
And using the reciprocal formula I found $u_n$ and $v_n$:
\begin{align} u_n & =\frac{n^x\cos(y\ln{n})}{n^{2x}\cos^2(y\ln{n})+n^{2x}\sin^2(y\ln{n})} \\ & = \frac{n^x\cos(y\ln{n})}{n^{2x}(\cos^2(y\ln{n})+\sin^2(y\ln{n}))} \\ & = \frac{n^x}{n^{2x}}\cos(y\ln{n}) \\ & = \frac{1}{n^x}\cos(y\ln{n}) \\ \end{align}
And similarly:
\begin{align} v_n & =\frac{-n^x\sin(y\ln{n})}{n^{2x}\cos^2(y\ln{n})+n^{2x}\sin^2(y\ln{n})} \\ & = \frac{-n^x\sin(y\ln{n})}{n^{2x}(\cos^2(y\ln{n})+\sin^2(y\ln{n}))} \\ & = -\frac{n^x}{n^{2x}}\sin(y\ln{n}) \\ & = -\frac{1}{n^x}\sin(y\ln{n}) \\ \end{align}
Now $\zeta(s)$ can be re-written as:
$$\zeta(s)=\zeta(x+yi)=\sum_{n=0}^\infty\left(\frac{1}{n^x}\cos(y\ln{n})-i\frac{1}{n^x}\sin(y\ln{n})\right)$$
Then I thought, the zeros of $\zeta(s)$ are when both the real and complex component equal to zero, so I used an online graphing tool to graph both:
$$\sum_{n=0}^\infty\frac{1}{n^x}\cos(y\ln{n})=0$$ and $$\sum_{n=0}^\infty-\frac{1}{n^x}\sin(y\ln{n})=0$$
This should show all the points where the sum is $0$. Then whenever both sums are $0$, it means $\zeta(s)=0$, and that point should be on both graphs.
Note: When graphing the sums, I obviously didn't sum up to $\infty$. I summed up to 100, as it seemed large enough.
These are the graphs:
The graph for the real component
The graph for the complex component
On top of each other, it looks like that:
Reminds the known plot:
The riemann zeta function $\zeta(s)$ plotted with domain coloring, taken from wikipedia
I expected to see the graphs colliding at the trivial zeros at least, but it doesn't seem like it. can anyone explain what these graphs actually mean?