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:**

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?

Prime Racessubject, and found out that it's mostly about prime numbers of the form $4n\pm1$ (not $6n\pm1$ as in my question). Have I missed anything? $\endgroup$ – barak manos May 12 '14 at 7:50