I forgot where it was, but I remember someone saying that you need $\phi(4)$, which is two, of Dirichlet's $L$ functions to get a prime counting function for primes of the form $4n\pm 1$ less than $x$.
During a quite minute today I thought the following might work as well:
$$ \int_0^x \delta\left(\sin\left(\pi\frac{n\pm 1}4\right)\right)\pi_0'(n)dn. $$
If I take $$ \pi_0'(x) = \operatorname{R}'(x) - \sum_{\rho}\operatorname{R}'(x^{\rho}) $$ with $\displaystyle \operatorname{R}'(x^k) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \frac{x^{k/n-1}}{\log x}$ and $\rho$ running over all the zeros of $\zeta$ function.
So only one $L$ function seems to be involved? Who is wrong?
with the same visual (embedding) effect (except in a comment here :-)).
– Raymond Manzoni
Jan 30 '13 at 12:56