I think in the last paragraph the generators of $G$ are meant to be the projections of the elements of $U$ to the first coordinate, and similarly the $x$ in the basis in $x^{f(x)} = 1$ should rather be its projection to the first coordinate -- or am I mistaken? Otherwise -- as far as I can tell!! -- this looks right to me. -- Thanks again!

@HW and Benjamin: I originally thought about asking which sets of primes can occur, but then preferred to ask a question that would more likely admit a definite answer -- and now you have answered even more than the former question. -- Great!!

@HW: Thank you very much! -- Now with your clarification regarding Higman's Embedding Theorem and the restriction to recursively enumerable sets of primes, it's clear.

In any case, for reasons of cardinality, the set of primes cannot be "whatever one wants" -- there are uncountably many sets of primes, but only countably many finitely presented groups.

Well -- but is the free product of infinitely many cyclic groups of prime order finitely generated? -- As I understand, Higman's Embedding Theorem is only for finitely generated groups, or am I missing something?

@Greg: well -- when running through the first 12596957371 primes (i.e. the primes up to about $3.206 \cdot 10^{11}$), the picture remains within the rectangle from -37501 to 1945 on the real axis and from -14390 to 16290 on the imaginary axis, while the square root of 12596957371 is about 112236. What about the constant(?) factor before the square root you expect? -- Can it be notably smaller than 1?

I don't know how to make this precise, and in particular how to exclude that what I observe are only small distortions coming from that by easy congruence conditions $p_i$ mod 8 and $p_{i+1}$ mod 8 are not statistically independent.

@Greg: As said, I know no further results on to what extent a simple random walk in $\mathbb{Z}^2$ is a suitable model here. However, key properties of the random walk are that its behaviour (i.e. how it continues) does neither depend on which point it has reached after a certain number of steps, nor on the number of steps it has already taken. I have computed a lot more pictures than those shown above, and it seems to me that the prime race in 2 dimensions doesn't share any of them. (continued in next comment)

@Greg: so you believe that the simple random walk in $\mathbb{Z}^2$ is a suitable model for the 2-dimensional prime race(?) -- Can you give reasons for this?

@Peter: Thank you very much for your discussion of the problem! So, to summarize, there are essentially two things which make the problem difficult: firstly the question of relations between $\pi(x;8,3) - \pi(x;8,7)$ and $\pi(x;8,1) - \pi(x;8,5)$, and secondly that information on exact numbers rather than just asymptotics is necessary.

I'm not a number theorist, but I don't think your question looks 'too elementary'. -- In any case you don't need to excuse for asking a question!

@Greg: I have added a plot of a random walk for comparison. I don't know any results on this, but it just looks different ... .