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I am trying to plot the identity seen here, namely that if we define: $$\psi _{0}(x)={\frac 12}\left(\sum _{{n\leq x}}\Lambda (n)+\sum _{{n<x}}\Lambda (n)\right)$$ Then, it equal to the following, where the sum is taken over the nontrivial zeta zeroes. $$\psi _{0}(x)=x-\sum _{{\rho }}{\frac {x^{{\rho }}}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\frac {1}{2}}\log(1-x^{{-2}})$$ The linked website show a very nice graphic where $n$ is the number of zeta zeroes used to approximate the function:

However, I am having trouble reproducing this. My goal is just to plot this for one value of $n$. So, my exact definition of the function in question seems to be working fine:

```
Ches0[x_] :=
1/2 (Sum[MangoldtLambda[n], {n, 1, x}] +
Sum[MangoldtLambda[n], {n, 1, x - 1}])
```

However, my implementation of the approximate identity, while it gets the values at the jumps right, seems to be very off in most other respects.

```
Ches0Appr[x_] :=
x - Log[2*Pi] - Sum[x^ZetaZero[i]/ZetaZero[i], {i, 1, 100}] -
1/2*Log[1 - 1/x^2]
```

I'm not exactly sure what's going wrong here, so my question is, how can I plot the second formula in my post?

Related. – corey979 – 2016-12-13T01:09:14.363