I don’t know much about the prime number theory and neither am aware of a lot of special mathematical functions. But am interested in having a general idea about how zeros of the Extended Harmonic Series are related to Prime Number distribution.

I was watching this video. It mentioned how non-trivial zeros of the extended Harmonic Series are related to the approximation of the modified version of the prime number counting function.

I figured out it is implemented as `PrimePi`

and I tried to make it look like the modified version as follows:

```
PrimePsi(x_):=Sum(If(PrimePowerQ@n,Log@Surd(n,PrimeOmega@n),0),{n,1,x});
```

But after some digging around I found a function `MangoldtLambda`

that does exactly that (which is surprisingly not on the function page of `PrimePi`

). So I modify as follows:

```
PrimePsi(x_):=Sum(MangoldtLambda(n),{n,1,x});
```

Now I can get the step function shown in the video as follows:

```
Plot(
{PrimePsi(x)},{x,0,32.5},
PlotLegends->"Expressions",
PlotRange->{{0,32.5},{0,34}},
ImageSize->Large,
AspectRatio->1/2
)
```

But how do I get that green function on the graph here? I know WL implements Extended Harmonic Series as `Zeta`

and its non-trivial zeros can be grabbed by `Im(ZetaZero(...))`

but what is that green function in the graph shown in the video and how do I make it in WL?