Series for $\text{Arg}( \zeta (z))$

Question

Though I don't know if the formula I've found is useful, I decided to publish it anyway. $$ \text{Arg}( \zeta (z)) = -\sum_ {k = 1}^{\infty}\sum _ {q = 1}^{\infty}\frac {1} {k P_q^{k x}}\text {Sin}( k y \text{ Log}(P_q ) ) \text{ , } |z|>1 $$

where $z=x+i y$ and $p_q$ is th $q^{th}$ prime number.

Here is an example with a plot of $\text{Arg}( \zeta (1+i y))$ in blue, and a plot of the sum in red:

My question is whether the formula is known, whether someone finds it helpful.

EDIT: this post was put 'on hold'. I don't know why. I'm a graphist and foreigner to the mathematics' community, thus it's very hard for me to check the results I got, and this is the reason of this post.

EDIT 2: If someone's interested, this other post shows at the end how the series was calculated simple tools to extract Re, Im, Abs... of any complex function

Answer 1

I have never seen anything like this before. Here is some code for the plot, apparently with a different scale factor:

(*Mathematica*)
Clear[a, k, q, y, x]
x = 1;
Monitor[a = 
   N[Table[-Sum[
       Sum[1/(k*Prime[q]^(k*x))*Sin[k*y*Log[Prime[q]]], {q, 1, 
         50}], {k, 1, 100}], {y, 1, 50, 1/4}]];, Floor[N[y]]]
g1 = ListLinePlot[a, DataRange -> {1, 50}, PlotStyle -> Red]
g2 = Plot[Arg[Zeta[1 + I*y]], {y, 1, 50}]
Show[g1, g2]