Fix $a, k \in \mathbb{N}$ relatively prime. For $x \in \mathbb{R}$ recall the function $$ \pi(x; k, a) = \sum_{\substack{p \leq x \\ p \equiv a \pmod{k}} } 1 $$ where $p$ denotes the primes. Chebyshev, Littlewood and Hardy, others, took an interest in studying the difference $$ \delta_\pi(x; 4,3,1) = \pi(x; 4, 3) - \pi(x; 4, 1). $$ Data suggests that $\delta_\pi(x; 4,3,1)$ is positive most of the time, meaning there seems to be more primes congruent to $3$ modulo $4$ than to $1$ modulo $4$. For example $\delta_\pi(x; 4,3,1)$ changes sign for the first time when $x = 26861$. Similarly $\delta_\pi(x; 3,2,1)$ changes sign for the first time when $x = 608981813029$. See here for instance http://arxiv.org/pdf/1202.3408v1.pdf
In at least the case when $k = 4$, although the sign first changes when $x = 26861$, we have that $\delta_\pi(x; 4,3,1) = 0$ for instance when $x = 5,6,17,18,41,42,461,462$. These are all the (8) zeros of $\delta_\pi(x; 4,3,1)$ for $x \in [3,1000]$. In the case when $k = 3$ however, my computer checks that $\delta_\pi(x; 3,2,1) \neq 0$ for $x \in [3,1000]$.
A small twist: For $N \in \mathbb{N}$ let $$ \rho(N; k,a) = \sum_{\substack{p \leq N \\ p \nmid N \\ p \equiv a \pmod{k}}} 1 $$ be the number of primes $p \leq N$ not dividing $N$ and in the arithmetic progression $a \pmod{k}$. Similarly consider $\delta_\rho$. From the computations I've done for $k = 3,4$ it seems that $\delta_\rho$ has many more zeros than $\delta_\pi$. For instance, there are 26 $N \in [3,1000]$ with $\delta_\rho(N; 3,2,1) = 0$. When $k = 4$, $\delta_\rho(N; 4, 3, 1)$ has 40 zeros in $[3,1000]$. It seems to me rather curious that such a small tweak can have such an effect on the balance between different residues with the same modulus. Could this possibly be of significance for the study of $\delta_\pi$? Also one could ask the following question:
Question: Given $k \geq 3$, are there infinitely many $N \in \mathbb{N}$ such that $\delta_\rho(N; k, -1, 1) = 0$?
I wonder if this has been considered before and if it might be of use to tackle problems related to $\delta_\pi$.