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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?

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    $\begingroup$ You are probably thinking at "There are $\phi(n)$ Dirichlet characters modulo $n$" here. Note that you didn't exhibit the Dirichlet character (or L-function) here (except if you think that it is $\zeta$...). Concerning the aspect of $\pi'(x)$ an animation is at Matthew Watkins link (bottom). $\endgroup$ – Raymond Manzoni Jan 30 '13 at 8:55
  • $\begingroup$ @Raymond I think it is $\zeta$, isn't it? Thanks for the animation... $\endgroup$ – draks ... Jan 30 '13 at 10:24
  • $\begingroup$ @RaymondManzoni would you mind if I use that gif (and a ref to the page, of course) here: Divergence of the Derivative of the Prime Counting Function $\endgroup$ – draks ... Jan 30 '13 at 12:44
  • $\begingroup$ I made these gifs for Matthew in 2000 or 2001 so they are kind-of his property. Anyway you may link the gif file directly with the ![xx](link) syntax !xx with the same visual (embedding) effect (except in a comment here :-)). $\endgroup$ – Raymond Manzoni Jan 30 '13 at 12:56
  • $\begingroup$ @RaymondManzoni Thx... $\endgroup$ – draks ... Jan 30 '13 at 12:59

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