I got these plots when I evaluate the sum of truncated $(k \bmod 4)$-Prime $\zeta$ function, i.e. $$ P_x(r,t)=P_{x;4,1}(-ir\cos t)+P_{x;4,3}(-ir\sin t)=\sum_{x\geq p\;\bmod\;4=3} p^{-ir\cos t}+\sum_{x\geq p\;\bmod\; 4=1} p^{-ir\sin t}. $$ To be precise the plots below ($r$ up, $t$ to the right) show $\displaystyle\log\left(\frac{|P_x(r,t)|}{\pi(x)}\right)$:

$\hskip0.6in$ enter image description here $\phantom{\Huge|}$ enter image description here $\phantom{\Huge|}$enter image description here $\phantom{\Huge|}$enter image description here $\phantom{\Huge|}$enter image description here $\hskip0.3in$$\scriptstyle \text{$r$ is plotted horizontal, $t$ vertical. Check the tooltips of the images for some infos and click to enlarge them. }$

Since I got the impression that the red spots appear more often, when the race between $4m\pm 1$ primes is tied, I preferred to use values from OEIS/A007351. I used the list of first 50 million primes from here.

Why are low values of $|P_x(r,t)|$ more frequent on the right hand side $\left(\frac\pi2\leq t \leq \pi\right)$?

The last two plots also use two equally large sets of primes, but the prime race is not tied there. It would be better to write them as $P_{x;4,1}(-ir\cos t)+P_{y;4,3}(-ir\sin t)$.

I would also be interested to find the values $r$ and $t$, resp. a functional dependency, where $P_x(r,t)$ is minimal. The red spots on the right already look like as they where lying on a line.

I collected some material of how to get analytical expressions about $P_{x;4,3}(r,t)$ here.



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