Why do the first spikes in these plots point in opposite directions?

Question

With the following Mathematica program:

Clear[x, n, nn]
nn = Prime[13] + 1;
A = -Sum[Re[x^N[ZetaZero[n]]], {n, 1, 50}];
Plot[A, {x, 0, nn}, PlotRange -> {-80, 120}]

I plotted the function:

$$f(x)=-\sum _{n=1}^{50} \Re\left(x^{\rho _n}\right)$$

Where $\rho _n$ is the $n$-th Riemann zeta zero.

With this second program:

Clear[n, k, x, nn]
nn = Prime[13] + 1;
Monitor[A = 
   Sum[Sum[Re[Log[Prime[n]]*x^(2 I \[Pi]*k/Log[Prime[n]])], {n, 1, 
      13}], {k, 1, 2*nn}];, k]
Plot[A, {x, 0, nn}, PlotRange -> {-200, 400}]

I plotted the function:

$$g(x) = \sum _{k=1}^{2 \text{nn}} \left(\sum _{n=1}^{13} \Re\left(\log (p_n) x^{\frac{2 i \pi k}{\log (p_n)}}\right)\right)$$

where $p_n$ is the $n$-th prime number.

In the first picture there is a spike at $x=1$ by the pointer, pointing down. In the second picture there is a spike at $x=1$ by the pointer, pointing up.

Is there a simple explanation to this? And can these first spikes in the plots be made to point in the same direction without changing the rest of the wave?

Answer 1

The spikes occur at the value $x=1$ and can be easily checked using exact arithmetic. For the first one, in fact, since $1^z=1$ for any $z$ we easily see that the value should be $-50$, as it appears in your graph. For the second, we get

Clear[n, k, x, nn]
nn = Prime[13] + 1;
A = Sum[Sum[
  Re[Log[Prime[n]]*x^(2 I \[Pi]*k/Log[Prime[n]])], {n, 1, 13}], {k, 1, 2*nn}];
A /. x -> 1

(* Out: 
  84 Log[2] + 84 Log[3] + 84 Log[5] + 84 Log[7] + 84 Log[11] + 
    84 Log[13] + 84 Log[17] + 84 Log[19] + 84 Log[23] + 84 Log[29] + 
    84 Log[31] + 84 Log[37] + 84 Log[41]
*)

N[%]

(* Out: 2801.31 *)

All looks OK to me.