Let $P$ be the prime zeta function

$$
P (s) = \sum_{p\, \in\text{ primes}} \frac 1 {p^s} = \frac {1} {2^s} + \frac {1} {3^s} + \frac 1 {5^s} + \frac 1 {7^s} + \frac 1 {11^s} + \cdots
$$

and define the semiprime zeta function as

$$
\zeta_{\Omega_2} (s) = \exp \sum _k^\infty \frac{P (k s)^2+P(2 k s)}{2 k}
$$

Then let $\gamma$ be the imaginary part the zeros of this function. When these then are plugged into 

$$
\sum_\gamma -\frac{\cos(\gamma \log t)}{\sqrt \gamma}
$$

the sum presumably then *should* peak at the semiprimes, (just as peaks are formed at the primes when summed over the imaginary parts of the Riemann Zeta function).

But the zeros are hard to find. I assumed that the zeros are along the line $ (\frac{1}{2}+it)$ *(a huge assumption, I know, given that the zeros of the prime zeta function are not on this line)*. I then calculated the near-zeros along this line with

    func[t_] := 1/Abs[Exp[(PrimeZetaP[(1/2 + I t)]^2 + PrimeZetaP[2 (1/2 + I t)])/2]];
    data = Table[{t, func[t]}, {t, .01, #, .01}] &[(**)10^4];
    peaks = Pick[data, PeakDetect[data[[;; , 2]], .01, .001], 1];

in Mathematica, which takes a long time to evaluate, so I uploaded the data here:

    peaks = ToExpression[Import["https://raw.githubusercontent.com/martinq321/peaks/main/omega2"]];

Below is a plot of the absolute value of the first term of $\zeta_{\Omega_2} (s)$

$\left| \exp \left(P \left (1/2+ i t \right)^2/2+P \left(1+ 2i t \right)/2\right)\right|$ with the red lines marking near-zero points:

[![enter image description here][1]][1]

Given the above, the plot below shows 
    
\begin{aligned}
\sum_{k \in \text{peaks}} -\frac{\cos (k \log t)}{\sqrt k}, \quad 200 \geq t \geq 250
\end{aligned}

where, despite the noise, the peaks show at the semiprimes and the primes:

[![enter image description here][2]][2]

and with the noise cleared up a bit:

[![enter image description here][3]][3]

    li2 = Select[peaks, #[[2]] > 5 &][[All, 1]]; x1 = 200; x2 = 250;
    data1 = Table[(Sum[-Cos[li2[[p]] Log[t]]/Sqrt@p, 
    {p, Length@li2}]), {t, x1, x2, .1}];
    ListLinePlot[BilateralFilter[data1, 1., 1, MaxIterations -> 25], 
    PlotStyle -> {AbsoluteThickness[3]}, DataRange -> {x1, x2}, 
    ImageSize -> 700, Frame -> True, AspectRatio -> 1/8]

What is going on here? I presume the peaks also showing at the primes because many of the 'zeros' are very close to the zeros of the zeta function. Is there a way to filter them out? Is there a better way to find the zeros of $\zeta_{\Omega_2} (s)$?


  [1]: https://i.sstatic.net/vh8pF.png
  [2]: https://i.sstatic.net/vP6Wz.png
  [3]: https://i.sstatic.net/uW236.png