Have not been able to get an answer to this on http://math.stackexchange.com, so trying here too...

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 $

Questions:

1. Has it been proved that $\forall k \exists n : k=|C^-(n)-C^+(n)|$?

2. Has any bound been proved for $|C^-(n)-C^+(n)|$ relatively to $n$ (e.g., $\ln \ln n$)?

3. What is the largest known value of $|C^-(n)-C^+(n)|$, and for what value of $n$ does it hold?