## “probabilistic” density of primes?

A certain set $\cal P$ of primes is defined by two assumedly independent conditions:

The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois extensions of $\Bbb Q$. The natural density of the bigger set ${\cal P}^\prime$ of primes satisfying this condition can be estimated under the usual independence hypotheses using Cebotarev's theorem.

Let $v$ be a non-zero vector in ${\Bbb Z}^2$. For each $p\in{\cal P}^\prime$ a certain $1$-dimensional subspace $V_p\subset({\Bbb Z}/p{\Bbb Z})^2$ is defined. The second condition is then that $p\in{\cal P}$ if and only if $v\bmod p\in V_p$.

Thus, a prime $p\in{\cal P}^\prime$ is actually in $\cal P$ with "probability" $c_p=\frac1{p+1}$.

I am tempted to consider the quantity $$\delta({\cal P})=\lim_{n\to\infty}\frac {\sum_{p\in{\cal P}^\prime_n}\ c_p} {|\hbox{primes \leq n}|}$$ where ${\cal P}^\prime_n={\cal P}^\prime\cap[1,\ldots,n]$. What is unclear to me is if this may be taken as a good estimate (or "guess") of the natural density of $\cal P$ or not.

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