Given the following two sets:
- $P^-(n) = \{p \leq n : p \equiv -1\pmod 6\}$
- $P^+(n) = \{p \leq n : p \equiv +1\pmod 6\}$
For example:
- $P^-(40) = \{5,11,17,23,29\}$
- $P^+(40) = \{7,13,19,31,37\}$
Given the following two functions:
- $C^-(n)=|P^-(n)|$
- $C^+(n)=|P^+(n)|$
For example:
- $C^-(40) = 5$
- $C^+(40) = 5$
Is there a known bound of the difference between $C^-(n)$ and $C^+(n)$?
In other words, is there a known maximum value for $|C^-(n)-C^+(n)|$?
Updated Question:
Has it been proved that $\forall k \exists n : k=|C^-(n)-C^+(n)|$?
Has any bound been proved for $|C^-(n)-C^+(n)|$ relatively to $n$ (e.g., $\ln \ln n$)?
What is the largest known value of $|C^-(n)-C^+(n)|$, and for what value of $n$ does it hold?